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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 revolutions 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 developed information models for forecasting technological progress. The forecasting trajectory of economic growth of developed countries (on the example of the USA) on the upward stage of the 6
^{th} LW (2018-2042), engendered by the digital technologies of the 4
^{th} Industrial Revolution is given. It is also demonstrated 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.

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 [

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 [

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 [

The crisis of world financial and economic system in 2008-2009, which led to the Great Recession in the USA and decrease in the majority of developed world economies, followed by the long global depression, lasting for almost 10 years, reminded the politicians, economists and businessmen that market economy 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 [

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 [

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 long-term 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 [

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 (1946-1982) 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” [

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 [

The Schumpeter-Kondratiev innovation and cycle theory was validated during the 5^{th} LW (1982-2018). The core for the 5^{th} technological paradigm was microelectronics, personal computers, information and communication technologies (ICT) and biotechnology [^{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 (1982-1994) 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 (1985-1995).

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 [

Then, with the help of the Hirooka innovation paradigm [^{th} technological paradigm and predicted the beginning of the 6^{th} LW—2017-2018. Next, we calculated economic potential of NBIC-technologies [^{th} long economic cycle (1982-1994), 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.

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 [

Y ( t ) = γ ⋅ K α ( t ) ⋅ ( A ( t ) ⋅ h ⋅ L ( t ) ) 1 − α + δ , (1)

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 ( δ > 0 ); γ is a constant normalizing index. It is explained by the fact that information— and 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:

Y ( t ) and L ( t ) [https://apps.bea.gov/iTable/iTable.cfm?reqid=19&step=2&isuri=1&1921=servey];

K ( t ) [https://apps.bea.gov/iTable/iTable.cfm?ReqID=10&isuri=1&step=1%20%20reqid%(1а) 3D10#reqid=10&step=1&isuri=1];

A ( t ) [https://fred.stloisfed.org/series/RTFPNAUSA632NRUG].

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 ( [

q Y = α ⋅ q K + ( 1 − α + δ ) ⋅ ( q A + q L ) , (2)

where

q Y = Y ˙ Y ; q K = K ˙ K ; q L = L ˙ L ; q A = A ˙ A . (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

a) Y ( t ) = Y 0 ⋅ exp [ ∫ T 0 t q Y ( τ ) d τ ] or b) A ( t ) = A 0 ⋅ exp [ ∫ T 0 t q A ( τ ) d τ ] . (3)

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.

PF (1) was verified for a number of developed countries of OECD and it turned out that it could function perfectly. For example, for the US economy from 1946 to 2018, we have received the following meanings for index γ and parameters α and δ in PF (1), basing on the range of factual values of main factors (Y, K, A and L), taken from the data base (1a) in corresponding prices, and using the method of least squares:

γ = 0.069 ; α = 0.622 ; δ = 0.167 . (4)

We took up in (1) h = 1 supposing that its real value will be based on the assessed value of normalizing multiplier/factor γ. In

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 [^{th} century there was a range of empirical regularities corresponding to the process of long economic growth, which are justified in a long-term 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 [^{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) K = σ K ⋅ Y , σ K = const ; b) Y = k K K , k K = c o n s t , (5)

where σ K is index of capital intensity; k K is the index of capital productivity; in this connection, σ K = k K − 1 ;

2) The shares of capital and labour in national income are nearly constant, i.e.

α = const and 1 − α + δ = const ; (6)

3) The wages for workers if labour share in GDP is constant grow in proportion to labour productivity i.e.

w ¯ ( t ) = a 0 ⋅ A ( t ) , (7)

where w ¯ ( t ) is a current average wage; a 0 is a normalizing index. Such wage growth was seen in the golden age of the capitalist economy flourishing (1948-1973).

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. First of all, Piketty convincingly showed that in the most developed countries (the USA, the UK, Germany, France, etc.) capital intensity ( σ K ) made a huge U-shaped curve in the 20^{th} century and returned to its maximum values at the beginning of the 21^{st} century, close to those observed at the end of the 19^{th} century ( [^{th}-19^{th} centuries the value σ K in the leading European economies was quite stable and amounted to σ K = 7 in France and the UK, σ K = 6.5 in Germany, and σ K = 4.5 in the USA ( [^{th} century the value σ K in European countries decreased to a minimum equal to σ K = 3.5 ÷ 3 , and in the USA it stopped at the level of σ K = 3.3 . 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 σ K 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 ( [^{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 ^{st} century around the stationary value

σ K ≅ 3.3 , ( k K = 0.3 ) . (8)

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 [^{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 ( [^{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).

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 forecasting of the main factors of economic growth—K, L and A. Let us start with capital accumulation. The growth of capital was the most important feature of capitalism in the 19^{th} and 20^{th} centuries. Piketty argues [^{st} century. Let us consider the patterns of accumulation of fixed physical capital in the 21^{st} century with the help of the classical equation of capital accumulation ( [

K ˙ ( t ) = I K ( t ) − μ K K ( t ) = s K Y ( t ) − μ K K ( t ) , (9)

where I K ( t ) is production investment; μ K is the rate of capital outflow; s K is accumulation rate. Here we are talking about the deterioration of infrastructure capital within the framework of one LW. For the US economy during the 5^{th} LW period (1982-2018), we have (1a):

μ K = 0.035 ; s K = 0.186 . (9а)

We estimated μ K value as the regression index in Equation (9) within the 5^{th} LW (1982-2018), and the s K 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:

K ˙ ( t ) = ( s k ⋅ k K − μ K ) K ( t ) . (10)

The solution for this equation is exponential capital growth:

K ( t ) = K 0 ⋅ exp [ ( s K ⋅ k K − μ K ) ( t − T 0 ) ] . (11)

If we take T 0 = 2018 as the beginning of the 6^{th} LW in the US economy, then we find in the World Bank database that K 0 = 59.3 trillion US dollars. Taking into account the specific values of the parameters k K (8), μ K , s K (9а), we find out that s K ⋅ k K − μ K ≅ 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 [

K ( t ) = K 1 + K 2 1 + u K ⋅ exp [ − ϑ K ⋅ ( t − T 0 ) ] , (12)

where K 1 , K 2 , u K and ϑ K are constant parameters; T 0 = 2018 . All these parameters are easily determined using the method of least squares only if the exponential trajectory of capital accumulation (11) coincides with the first half of the logistic curve (12), up to the inflection point (2018-2034), which is located in the middle of the 6^{th} LW (2018-2050). Hence, the following estimates are obtained as a result of the calculations for the US economy:

K 1 = 48.46 trillion dollars; K 1 = 105 trillion dollars; u K = 5 ; ϑ K = 0.1 . (12а)

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 h ≡ 1 there. If we now assume that the first Kaldor regularity (5) will be fulfilled, which means balanced economic growth, when q Y = q K , then, substituting relation (5в) into PF (1), we obtain the equation for determining the effective amount of labour:

A L = ( k K γ ) 1 1 − α + δ ⋅ K 1 − α 1 − α + δ . (13)

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:

L p t ( t ) = ( k K γ ) 1 1 − α + δ ⋅ 1 A ( t ) ⋅ K 1 − α 1 − α + δ ( t ) . (14)

Thus, according to Formulas (13) and (14), we can carry out a predictive calculation of the growth trajectory AL (13), since the predicted trajectory of movement K ( t ) is already known and is calculated by Formula (12), or of the predicted trajectory of growth of potential workplaces L p t ( t ) (14) if A ( t ) is known (see

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 ( [

L p e ( t ) = L b d + λ ( t − T b d ) , (15)

where L b d is the number of employed (workplaces) in the economy in the initial year ( T b d = 2018 ) of the rise of the 6^{th} Kondratiev LW (2018-2050); index λ is determined on the condition that in 2030 the number of employed will increase by 2 million workers.

That is why for the US economy, where L b d = 136.6 million workers, λ = 0.17 . The graph of employment growth L_{1} in the US economy in the 2020s and 2030s, calculated with the help of the empirical forecast Formula (15), is shown in ^{th} Kondratiev LW (1982-2018) (see _{2}, calculated using Formula (14) and built on the same

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 Y = k 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

_{1} and q_{2} at the upward stage (2018-2042) of the 6^{th} LW (2018-2050), calculated according to the

abovementioned forecast trajectories Y_{1} and Y_{2}. As can be seen from the graphs q 1 ( t ) and q 2 ( t ) , they make up for about 2% per year in the next 15 years, as well as in the years of depression (2010-2017), which is directly seen in

Above, we forecasted long-term balanced economic growth using the well-known 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 A d ( 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 ( [

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.

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) ( [

q A d ( t ) = ξ ⋅ I d K d ⋅ S ˙ A d S A d = ξ ⋅ ε d ( t ) ⋅ q s ( t ) , (16)

where I d ( t ) is current investment in physical capital K d ( t ) of the information and digital economy; S A d ( t ) is a function describing the dynamics of the accumulation of industrial technological knowledge, which determines the advanced technological level in the economy A d ( t ) ; ε d ( t ) = I d / K d is a relative level of investment in the economy; q S ( t ) = S ˙ A d / S A d is the growth rate of production technological information in the economy; ξ is a constant calibration index.

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 [

q A d ( t ) = A ˙ d ( t ) A d ( t ) = ξ I d K d ⋅ S ˙ A d S A d = ξ ⋅ ε d ( t ) ⋅ q S ( t ) . (17)

Since information and digital technologies are exponentially growing, it can be assumed that S A d ( t ) will also be an exponential function. Indeed, we have defined (for more details see [_{Ad} function is an exponential function written as

S A d ( t ) = S A d 0 exp [ g ( t ) ] . (18)

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 ( [_{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,

S ˙ A d ( t ) = S A d 0 ⋅ g ˙ ( t ) ⋅ exp g ( t ) . (19)

If we substitute representations (19) and (18) into Formula (17), we get a compact formula for calculating the required growth rates of technological progress:

q A d ( t ) = ξ ε d ( t ) g ˙ ( t ) . (20)

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

q S A d ( t ) = S ˙ A d ( t ) S A d ( t ) = g ˙ ( t ) . (21)

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) that

A d ( t ) = A d 0 exp [ ∫ 0 t q A d ( τ ) d τ ] . (22)

If we put here the expression for q_{Ad} (20), we will finally have:

A d ( t ) = A d 0 exp [ ξ ∫ 0 t ε d ( τ ) g ˙ ( τ ) d τ ] . (23)

In the work [

In the mathematical form this means that there exists a certain Lagrangian, the functional of which, in accordance with the Klimontovich principle, takes an extreme value at the upward stage of the Kondratiev LW. The corresponding Lagrange equation generates the required dynamical regime of production of technological information. Let us give here four dynamical regimes of production of technological information that are of greatest interest from the point of view of describing information retrospective and forecasting the digital future.

Information is produced at a constant growth rate g ˙ = v 0 = const . The Lagrange function has the form of

L ( g ˙ , g , t ) = g ˙ 2 . (24)

The corresponding Lagrange equation

d d t ∂ L ∂ g ˙ − ∂ L ∂ g = g ¨ = 0 (24a)

has the solution

g ( t ) = g 0 + v 0 t , g ˙ 0 = v 0 . (24b)

In this case, S d ( t ) = S d 0 exp ( v 0 t ) . 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).

Growth rates of information production grow exponentially with its accumulation ( [

L ( g ˙ , g , t ) = g ˙ 2 e − 2 g . (25)

The corresponding Lagrange equation has the following form: g ¨ = g ˙ 2 . The solution of this equation under initial conditions g ( t = 0 ) = g 0 and g ˙ ( t = 0 ) = v 0 leads to a hyperbolic increase in growth rates of information production:

g ˙ ( t ) = 1 T S − t ; g ( t ) = g 0 − ln ( 1 − t T S ) ; T S = 1 v 0 , (25a)

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 [_{S} = 2026. In reality, the explosive demographic growth was replaced with the stabilization dynamical regime—demographic transition [^{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.

This regime is a combination of the dynamical regime with aggravation g ˙ - e g , which is implemented at the initial stage of development, and the invariable dynamical regime g ˙ = const at the final stage: g ˙ ~ e g 1 + e g .

The Lagrangian has the form of

L ( g ˙ , g , t ) = g ˙ 2 e − 2 g 1 − g ˙ , (26)

and Lagrange equation

g ¨ = g ˙ 2 ( 1 − g ) . (26a)

Usually the solution needs to be scaled to make it adequate for the problem at hand. This is done by introducing new variables

g = g ¯ s g ; t = t ¯ s t , (26b)

where s g and s t are scaled factors.

The scaled solution of the latter Lagrange equation has the form:

а) g ˙ ( t ) = 1 s g [ 1 + c 1 e − s g g ( t ) ] − 1 ; c 1 = e s g g 0 ( 1 v 0 − 1 ) ; (27)

b) s t ⋅ t = s g g ( t ) − c 1 e − s g g ( t ) + c 2 ; c 2 = 1 v 0 − 1 − s g g 0 .

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 _{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 [^{2} = 0.998).

In this scenario, at the initial stage, the process will be sharply escalated ( g ˙ - e g ) and, due to inertia, it skips the stationary mode ( g ˙ = const ), and then, having reached a certain maximum value g ˙ m , it returns asymptotically to the stable

level. This regime can be described by the relation g ˙ ~ e g 1 + z ( g ) e g , where z ( g ) is the deceleration function, which in the simplest case has the form z ( g ) = 1 − 1 1 − ρ e − ρ g , where ρ = const and ρ ≠ 1 , and if ρ → 0 the dynamical

regime with aggravation is obtained, while if ρ → ∞ , it gets to the dynamical regime with stabilization. This regime is produced by the Lagrangian

L ( g ˙ , g , t ) = g ˙ 2 e − 2 g 1 − z ( g ) g ˙ , and the corresponding Lagrange equation has the form of

g ¨ = g ˙ 2 { 1 − g ˙ [ z ( g ) + d z d g ] } . (28a)

Scaled solution of such an equation has the following form:

a) g ˙ ( t ) = 1 s g [ 1 − e − ρ s g g 1 − ρ + c 1 e − s g g ] − 1 ; c 1 = e s g g 1 ( 1 v 1 − 1 + e − s g g 1 1 − ρ ) ; (28)

b) s t ⋅ t = s g g + e − ρ s g g ρ ( 1 − ρ ) − c 1 e − s g g + c 2 ; c 2 = 1 v 1 − 1 − s g g 1 − 1 ρ e − ρ s g g 1 .

Here v 1 = g ˙ ( t = 2018 ) and g 1 = g ( t = 2018 ) 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 [

Graphs of growth rates of technological information v ( t ) = g ˙ ( t ) described in Equations (27a) and (28a) are shown in ^{th} informational LW (1982-2018), we will show that curve v 1 ( t ) allows us to accurately describe the technological progress in the developed economy.

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):

q A d ( t ) = ξ ε d ( t ) ⋅ g ˙ ( t ) , (29)

where g ˙ ( t ) is the production rates of technological information in the economy; ε d ( t ) is relative investments of I d ( t ) in the fixed productive capital K d ( t ) of the information and digital economy. Here and in what follows, the normalizing index ξ is assumed to be equal to unity ( ξ = 1 ), 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 (1982-2018), 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 q ¯ A d ( 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

ε d ( t ) = ε 0 + ε 1 ( t − T 0 ) , (30)

where T 0 = 1982 ; ε 0 = 0.09 ; ε 1 = 0.002 . To calculate g ˙ ( t ) in the previous paragraph, (§2.5) the following equations were obtained (27):

а) v ( t ) = g ˙ ( t ) = 1 s g [ 1 + c 1 ⋅ e − s g ⋅ g ( t ) ] − 1 ; c 1 = ( 1 s g v 0 − 1 ) e s g g 0 ; (31)

b) t = 1 s t [ s g ⋅ g ( t ) − c 1 ⋅ e − s g ⋅ g ( t ) + c 2 ] ; c 2 = 1 s g v 0 − 1 − s g g 0 .

Let us introduce scaled values:

t ¯ = s t ⋅ t ; g ¯ ( t ) = s g ⋅ g ( t ) ; g ¯ ˙ ( t ) = s g ⋅ g ˙ ( t ) . (31a)

First of all, it is necessary to determine the initial values g 0 and v 0 , characterizing the production of technological information. To determine v 0 , we use the hyperbolic equation for the growth of rates of information production during the initial period of information technology development (25d), according to which the point of singularity is T s = v 0 − 1 . It is natural to take 1946 as the beginning of the information age, when the world’s first universal computer was put into operation. The explosive growth of the influence of information and communication technologies (ICT) on the economic development of leading countries was first observed in 1995-1997. Therefore, T s ≅ 50 years and v ′ 0 = 0.02 at the initial stage. Yet g_{0} is determined from the solution (25b), assuming that g ′ 0 = 0 for 1946. Then, for the initial year of the upward stage of the 5^{th} LW (1982), from (25b) we get:

g ¯ 0 = ln ( 1 − 36 50 ) ≅ 1.273 . (32)

Next, according to the fact that the logistic function (31a), which describes the production rate of technological information, takes at the beginning of the 5^{th} LW (1982) and at the upper turning point (2004, see

а) T r = 2004 ; v ¯ r = g ¯ ˙ r = 0.9 = { 1 + ( 1 v 0 − 1 ) e − ( g ¯ r − g ¯ 0 ) } − 1 ; (33)

b) T 0 = 1982 ; v ¯ 0 = g ¯ ˙ 0 = 0.1 = { 1 + ( 1 v 0 − 1 ) e − ( g ¯ r − g ¯ 0 ) } − 1 .

It follows from the second equation that if

T 0 = 1982 , v ¯ 0 = 0.1 . (33c)

From the first equation we get:

T r = 2004 ; g ¯ r = 5.67 . (33d)

Knowing v ¯ 0 = 0.1 and g ¯ r = 5.67 , we can calculate numerical values of integration constants c 1 (31a) and c 2 (31b):

c 1 = 32.14 ; c 2 = 7.73 . (34)

Now let us move on to determining the numerical values of the scaled indexes s t and s g (3с). Substituting the expressions for c 1 (31а) and c 2 (31b), into Equation (31b), we get the following equation at the upper turning point of the upward wave ( T r = 2004 ) of the 5^{th} LW:

s t = 1 T r − T 0 [ g ¯ r − g ¯ 0 + ( 1 v ¯ 0 − 1 ) e − ( g ¯ r − g ¯ 0 ) + 1 v 0 − 1 ] . (35)

Substituting here numerical values T r , T 0 , g ¯ 0 (32) and g ¯ r (33d), we get

s t = 0.6 . (36)

Substituting the expression for the production rate of technological information g ˙ ( t ) (31а) and (31b) in Formula (29) we get:

q A d ( 1 ) ( t ) = ε d ( t ) s g [ 1 + g ¯ ( t ) − s t ⋅ t + c 2 ] , (37)

where t = T − T 0 ; T 0 = 1982 ; T 0 ≤ T ≤ T r = 2004 . Moreover, here function g ¯ ( t ) is in a numerical form by solving the nonlinear Equation (31b) at each point t from a given timespan (37):

s t ⋅ t = g ¯ ( t ) − c 1 ⋅ e − g ¯ ( t ) + c 2 , (38)

where g ¯ 0 ≤ g ¯ ( t ) ≤ g ¯ r .

Since the left side of Equation (37) is set by its actual values, which have already been presented in

s g = 412.33 . (39)

Now we can build a trend trajectory of the rates of technological progress by calculating q A d ( t ) on the right side of Formula (37) with the concrete value s g = 412.33 (39). The trend trajectory of the rates of technological progress q A d ( t ) at the upward stage of the 5^{th} LW (1982-2004) is shown in

At the downward stage of the 5^{th} LW (2004-2018), to approximate the trajectory of technological progress, we shall first use the formula for deceleration in technological progress in the recession phase, obtained in the work [

q A d ( 2 ) ( t ) = q A d r ( 1 ) ⋅ exp { − [ 1 − λ 0 ( t − T r + 1 λ 0 ) e − λ 0 ( t − T r ) ] } , (40)

where q A d r ( 1 ) is the value of q A d ( 1 ) ( t ) (37) at the upper turning point ( t = T r ) T r = 2004 . Using the actual data q ¯ A d ( t ) given in ^{th} LW (2004-2014) with the help of the method of least squares we get:

λ 0 = 2.73 . (40а)

The corresponding part of the trend trajectory of technological progress in the phases of recession and depression is also shown in ^{th} LW, which is best approximated by an exponential curve:

q A d ( 3 ) ( t ) = q A d e ( 2 ) exp [ ω ( t − T r e ) ] , (41)

where q A d e ( 2 ) is the value q A d ( 2 ) (40) in the lower turning point ( T r e = 2014 ).

Proceeding from actual data q ¯ A d ( t ) in the recovery phase (2014-2018, see

ω ≅ 0 . (41a)

Thus, we approximated the rates of technological progress q A d ( t ) of the 5^{th} LW both at the upward (37) and downward (40) and (41) stages. Now we can calculate the trajectories of both actual and model technological progress A ¯ d ( t ) and A d ( t ) by the formula

A d ( t ) = A d 0 ⋅ exp [ ∫ T 0 T q A d ( t ) d t ] . (42)

The initial value A d 0 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

A d 0 = 1 L 0 ( Y 0 γ K 0 α ) 1 1 − α + δ . (43)

The trajectory of the actual technological progress is obtained by numerical integration (42), using the actual data q ¯ A d ( t ) presented in

Now let us consider how this model error can affect the further calculations of the trajectory of economic growth by the production function (1):

Y ( t ) = γ K α ( t ) [ A ( t ) ⋅ L ( t ) ] 1 − α + δ . (44)

Here we have just calculated and verified A d ( t ) with high accuracy, with a mean square error not exceeding 2%, over a time span of 36 years. The actual data for the main economic variables Y ( t ) , K ( t ) and L ( t ) for the US economy were taken from sources (1a). First, the optimal values of the production function parameters (44) γ, α and δ were calculated:

a) At current prices: γ = 0.04 ; α = 0.73 ; δ = 0.33 (45)

b) At comparable prices: γ = 0.07 ; α = 0.63 ; δ = 0.19 .

After having calculated these both variants, we obtained a mean square error at current prices σ Y ( 1 ) = 0.3 % and at comparable prices σ Y ( 2 ) = 0.13 % . The trajectories of economic growth are shown in

As can be seen from the graphs in

In conclusion, let us calculate the growth trajectory of the production of technological information g ( t ) from Formula (40):

g ( t ) = g 0 + s t ⋅ ∫ T 0 T q A d 2 ( t ) ε d ( t ) d t , T 0 ≤ T ≤ T b d ; (46)

T 0 = 1982 ; T b d = 2018 ; g 0 = g ¯ 0 s g ; g ¯ 0 = 1.27 ; s g = 412.33 ; s t = 0.6 .

Here, in the integrand, we successively use approximating functions (37), (40) and (41). As a result of calculations, we get the trajectory g ( t ) shown in

g b d ≅ 0.016 or g ¯ b d ≅ 6.6 . (46a)

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):

а) ν ¯ ( t ) = g ¯ ˙ ( t ) = [ 1 − e − ρ g ¯ ( t ) 1 − ρ + c 3 ⋅ e − g ¯ ( t ) ] − 1 ; c 3 = e g ¯ b d ( 1 ν ¯ b d − 1 + e − ρ g ¯ b d 1 − ρ ) ; (47)

b) t ¯ = g ¯ ( t ) + e − ρ g ¯ ( t ) ρ ( 1 − ρ ) − c 3 ⋅ e − g ¯ ( t ) + c 4 ; c 4 = 1 ν ¯ b d − 1 − g ¯ b d − e ρ g ¯ b d ρ .

Here g ¯ ( t ) = s g ⋅ g ( t ) ; t ¯ = s t ⋅ t ; g ¯ b d = g ¯ ( t = T b d ) ; T b d = 2018 ; g ¯ b d ≅ 6.6 (46a); ρ = 0.2 . The scaled factors s t and s g have already been defined for the information age: s t = 0.6 (36) and s g = 412.33 (39). They also retain their values in the digital age. Taking into account that at the initial point of the upward stage of the 6^{th} LW ( T b d = 2018 ), the value of the logistic function ν ¯ ( t ) (47a) should equal ν ¯ ( t ) (47a) and solving Equation (47a) at this point we make sure that the latter turns into identical equation. Therefore,

ν ¯ b d = 0.1 . (48а)

Then we can easily find numerical values of integration constants:

c 3 ≅ 6559.3 ; c 4 ≅ 1.1 . (48b)

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) at the point of return (return to) t = T r t is reduced to a simplified equation:

ν ¯ r t = g ¯ ˙ r t = 1.1 = [ 1 − e − ρ g ¯ r t 1 − ρ + ( 1 ν ¯ b d − 1 + e − ρ g ¯ b d 1 − ρ ) e − ( g ¯ r t − g ¯ b d ) ] − 1 . (49)

Here all values except for g ¯ r t are known. Solving the equation, we find out that

g ¯ r t ≅ 12.72 . (49a)

To find the duration of the upward stage of the 6^{th} Kondratiev LW t r t = T r t − T b d , we use Equation (47b) at the upper turning point ( t = T r t ):

t ¯ r t = g ¯ r t + e − ρ g ¯ r t ρ ( 1 − ρ ) − c 3 ⋅ e − g ¯ r t + c 4 ≅ 14.3 .

Since the scaled factor is s t = 0.6 (36), then t r t = t ¯ r t s t ≅ 23.8 years . Therefore, we get

t r t ≅ 24 years and T r t = 204 . (49b)

Thus, the duration of the diffusion of digital technologies into the economy is standard for innovative technologies and consists of 24 years. Solving Equation (47b) with the help of the numerical method with respect to g ¯ ( t ) in the range T b d ≤ t ≤ T r t , we obtain the numerical values of the function g ¯ ( t ) , the growth trajectory of which is shown in

After having determined all the constant parameters and indexes of function g ˙ ( t ) (47), describing the production rates of technological information, we can turn to calculating the predicted trajectory of the growth rates of technological progress q A d ( t ) (29). Substituting the expression for g ˙ ( t ) (47) into the original Formula (29) and admitting ξ = 1 , we obtain the following formula for predictive calculations q A d ( t ) :

q A d ( t ) = ε d ( t ) s g ⋅ [ 1 + c 4 + g ¯ ( t ) + 1 ρ e − ρ g ¯ ( t ) − s t ⋅ t ] , (50)

where t = T − T b d , T b d = 2018 , T b d ≤ T ≤ T r t = 2042 ; ε d ( t ) = ε 0 + ε 1 ( T − T 0 ) , T 0 = 1982 , ε 0 = 0.09 , ε 1 = 0.002 . The forecasted values of the function g ¯ ( t ) have already been calculated and presented in a graphical form in ^{th} Kondratiev LW (2018-2042) for the US economy, are presented in a graphical form in

A d ( t ) = A b d ⋅ exp [ ∫ T b d T q A d ( t ) d t ] . (51)

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):

Y ( t ) = γ K α ( t ) [ A d ( t ) L p ( t ) ] 1 − α + δ (52)

The predicted growth trajectory of technological progress in the digital age A d ( t ) has already been calculated above (51). Capital accumulation K ( t ) , as was shown earlier, is forecasted by Formulas (11) or (12). Let us choose the simplest exponential law of capital accumulation (11):

K = K b d ⋅ exp [ 0.021 ( T − T b d ) ] , (53)

where K b d = 59.3 trillion US dollars; T b d = 2018 .

Employment dynamics is forecasted using Formulas (14) or (15). Let us choose the predictive Formula (15) based on an empirical pattern:

L p e ( t ) = L b d + λ ( T − T b d ) , (54)

where L b d = 136.6 million workers; λ = 0.17 . Constant parameters α and δ and a normalized index γ have already been given (45b).

In _{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”.

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 [

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

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 countries— their 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.

As can be seen in

Labour productivity in the symbiosis “human + IM” A h ( t ) in the age of the digital economy can be described by the Equation ( [

A ˙ h ( t ) = q h ( t ) A d β ( t ) ⋅ A h 1 − β ( t ) , (55)

where A d ( t ) is digital technological progress (maximum level of world digital technologies); q h ( 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:

q h ( t ) = q h m 1 + η ⋅ exp [ − ϑ ( t − T b d ) ] (56)

If we accept that the duration of a business cycle is 8 years, then the following estimates are easily obtained for parameters η and ϑ:

η = 9 ; ϑ = 0.55 . (56a)

q h m will be estimated later.

Let us consider differential Equation (55) in greater detail. Parameter β in this equation characterizes the part of the work that is automated and given to IM, while ( 1 − β ) is the remaining part of the work that is fulfilled by the human. If β = 0 , we obtain the simplest equation from Equation (55)

A ˙ h ( t ) = q h ( t ) A h ( t ) , (57)

which describes the growth of employee productivity without the use of digital technologies and IM. This equation is identical to the one describing the accumulation of human capital h ( t ) during the years of work after graduation ( [

h ˙ ( t ) = q h ( t ) ⋅ h ( t ) . (58)

After graduation a person has some knowledge described by the formula ( [

h = exp ( ψ ⋅ u ) , (59)

where u is an average number of years of study; ψ is the index of return on education. Empirical estimates of this index have shown that 0.06 ≤ ψ ≤ 0.1 ( [

q h m = ψ = 0.06 . (60)

When β = 1 , then Equation (55) will be:

A ˙ h ( t ) = q h ( t ) A d ( t ) . (61)

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. For any other values of 0 < β < 1 , Equation (55) shows that labour productivity in the economy grows in proportion to the average weighted quantity of employee’s productivity and the level of digital technological progress. In what follows, we shall consider three values of β:

β 1 = 1 3 ; β 2 = 1 2 and β 3 = 2 3 (62)

In the first case fragments of work performed by a person prevail, in the third— by 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:

A h ( t ) = [ A h 0 β + γ ∫ T b d T q h ( t ) A d β ( t ) d t ] 1 β . (63)

Since the employee’s labour productivity at the initial moment A h 0 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. u = 18 ). Suppose it took an employee 3 years with the highest return ( ψ = 0.1 ) to fully master the knowledge and skills of working with IM. Then u = 21 and if ψ = 0.1 , by Formula (59) we get

A h 0 = 8.17 . (64)

Thus, we have obtained Formula (63), which allows us to calculate the dynamics of labour productivity growth in the digital economy for any values of the parameter β.

In practice, for a comparative analysis, it is not the trajectories of labour productivity growth (63) that are more suitable, but their growth rates q A h ( t ) , which can be most easily obtained from Equation (55):

q A h ( t ) = A ˙ h ( t ) A h ( t ) = q h ( t ) ( A d ( t ) A h ( t ) ) β . (65)

Substituting the expressions for A d ( t ) (51), A h ( t ) (63) and q h ( t ) in Formula (56), we get:

q A h ( t ) = q h ( t ) ⋅ A d β ( t ) A h 0 β + γ ∫ T b d T q h ( t ) A d β ( t ) d t

where

q h ( t ) = q h m 1 + η ⋅ exp [ − ϑ ( t − T b d ) ] . (66)

Here A h 0 = 8.17 (64); η = 9 , ϑ = 0.55 (56a); q h m = 0.06 (60).

The growth rates of labour productivity in the digital economy with the widespread and effective use of the symbiosis “human + IM”, calculated by Formula (66), are presented graphically in

β 1 = 1 3 ; β 2 = 1 2 ; β 3 = 2 3 . The corresponding curves of labour productivity

growth in the digital economy are indicated by the symbols q_{1}, q_{2} and q_{3} (see

As is seen in the growth curves of labour productivity in

and the lowest labour productivity is observed at β = 2 3 , 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

If we now use the predictive Formulas (52)-(54) to calculate the trajectory of GDP in the digital era (2018-2042), replacing A d ( t ) in the production function (52) by A h ( t ) (63), we get graphs Y_{1}, Y_{2} and Y_{3}, shown in _{1}, q_{2} and q_{3}, which are shown in _{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.

1) For long-term forecasting of technological progress and economic growth, it is more suitable to use the Schumpeter-Kondratiev innovation and cycle theory on the formation of long waves (LW) of economic development, which lasts for about 30 years under the influence of a powerful cluster of innovative technologies generated by cyclically arising industrial revolutions. The Solow neoclassical economic growth model, tied to the LW, makes it possible to accurately predict the economic dynamics of technologically developed countries with the longest forecasting horizon of up to 30 years. Currently, leading countries have entered the upward phase of the 6^{th} LW (2018-2042) under the influence of digital technologies of the 4^{th} Industrial Revolution.

2) In the information and digital age, technological progress plays a key role among the main factors of economic growth (capital, labour and technological progress). The authors have developed an information model which allows for forecasting technological progress basing on growth rates of endogenous technological information in the economy. The main dynamical regimes of producing technological information corresponding to the eras of information and digital economies are highlighted, and the Lagrangians that generate them are constructed. The information model of technological progress was verified on the example of the 5^{th} information LW in the economies of developed countries (1982-2018) for the US economy and was proven highly accurate. High accuracy also occurs when predicting the trajectory of economic growth using an information model for calculating technological progress. It is important to note that forecasts of the accumulation of productive capital and the dynamics of the number of employed in the economy are carried out according to the classical models.

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.

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.

The authors declare no conflicts of interest regarding the publication of this paper.

Akaev, A. and Sadovnichiy, V. (2021) Information Models for Forecasting Nonlinear Economic Dynamics in the Digital Era. Applied Mathematics, 12, 171-208. https://doi.org/10.4236/am.2021.123012