Research on the mechanism of computing power resources to promote the development of digital economy-based on the perspective of new quality productivity

1)

Cobb-Douglas production function.Computing power resources are the “new energy” in the era of digital economy, which have economic attributes like water, electricity and other energy sources. Many scholars at home and abroad use the Cobb-Douglas production function (referred to as the C-D function) to study the relationship between energy consumption and economic growth[19]This also provides a reference for the research of computing power resources. The C-D function is simple in form, easy to estimate and interpret, and suitable for use in preliminary analysis and model construction. Even for power research, it can effectively capture the relationship between computing power and other factors of production, and provide clear economic explanation and empirical support. Especially, when the data is limited, the C-D function provides a flexible and efficient modeling tool. C-D function, its general form is: Q = A * K ^α * L ^β, is an economic mathematical model analyzing the production path of the industrial system of a country or region, Q, K, L respectively represent output value, capital, labor force, A is A constant parameter representing technical elements, α is the elasticity coefficient of capital K, β is the elasticity coefficient of labor force L. Logarithmic: obtain ln (Q) = ln (A) + α ln (K) + β ln (L), and conduct regression calculation to observe the impact of capital and labor input on output, and investigate the α + β situation.

ρ Is the resolution coefficient, the values range between 0 and 1, in this study, ρ was taken as 0.5. (6)

calculate the grey correlation degree (2)ζi=1nk=1nζi(k).$$\zeta {\rm i} = 1{\rm nk} = 1{\rm n}\zeta{\rm i} ({\rm k}).$$

Using the gray correlation analysis formula, the correlation coefficient between each comparison sequence and the reference sequence is calculated to see whether there is a significant positive correlation between the discrete data. If the correlation coefficient is greater than 0.5 indicates that the data is selected correctly, it can be inserted into the C-D function for calculation.

Hypothesis 1: into the digital economy era technology progress, is closely related to time, so the Dutch economist James, improved C-D function computing power research, improve mainly increase the time parameters, the equation for (3)Q=A0e^λt∗K^α∗L^β^λ$${\rm Q}={\rm A}0{\rm e} \hat{}\lambda {\rm t} \ast {\rm K} \hat{} \alpha \ast {\rm L} \hat{}\beta \hat{}\lambda$$

A0e ^λ t represents t period technology progress level, A0 for the initial technical level, λ, K represents capital input, L represents human input, α, β is the elastic coefficient of capital and human input. Logarithmic processing: get ln (Q) = ln (A) + λ t + α ln (K) + β ln (L), and use ptyhon software for regression calculation to observe the impact of technical innovation, capital and labor input on the computing power scale, and the remuneration of model scale.

As the base of the digital economy, there is a production relationship between the computing power scale output Q and the capital input K (computer, communication and other electronic equipment manufacturing input) and the human input L (the number of employees in information transmission, software and information technology services). Let Q be the reference sequence, A0e λ t, K and L be the comparison sequence, investigate the data relevance, and substitute the productivity function to calculate and verify the interrelationship.

Hypothesis 2: economic perspective, force become new quality productivity instead of human effect on the development of digital economy, C-D function no longer the input-output but the contribution rate of production factors, the equation for Y = AK α Q β, formula: Y total output, K material capital input, α material capital elastic coefficient, A production technology level, labor with force input Q, β force input elastic coefficient. Output Y and capital input K are all measured in one trillion yuan, ignoring the depreciation of fixed assets. Since the model contains parameters that are nonlinear, natural log transformation of both sides of the equality to form a multivariate linear equation, lnY = lnA + α lnK + γ lnQ

The gray correlation model is used to process data. As the computing power resources is a new type of infrastructure, the statistics time is not long. The original data of 2018-2022, such as the China Digital Economy Development Index report, the White paper of China Computing Power Development Index, China Statistical Yearbook and other literature materials are obtained as follows:

The data are from the Ministry of Industry and Information Technology, China Academy of Information and Communications Technology, Kezhi Consulting, and National Bureau of Statistics.The grey correlation between the variables was calculated as follows:

According to the calculation results, all the correlation coefficients exceeded 0.5, indicating the accurate variable selection and positive correlation, which can be substituinto the productivity function to further determine the correlation. Fixed asset investment K has an important impact on the output of computing power scale (Q). The progress level of technology over time A0e λ t is also highly associated with computing power scale (Q), and the impact of human input is small. From digital industrialization, digital industry as sequence parameter calculation found that calculate force as a human, the impact of both is almost the same (about 0.65), the impact of capital industry digital than digital industrialization (0.967,0.936), which can also explain the industry digital GDP five years from 24.88 trillion yuan rose 1.65 times to 41 trillion.

1)

Regression analysis on Hypothesis 1

It is crucial to use the raw data with practical significance in the economically modified Cobb-Douglas production function Q=A0e^λ t * K^α * L^β analysis. The data set includes computing power scale (Q), technological progress level (A), investment increase in computer and electronic equipment manufacturing (K), and the number of employees in information technology services (L). These data are evaluated by experts for parameter estimation and economic analysis of C-D function, including calculation force scale (Q) as dependent variable, and investment K and human L as independent variables. Take the natural log of both sides of the production function and convert it into linear form: ln (Q) = ln (A0) + λt + αln (K) + βln (L). Regression analysis was performed using the Python:

First, the model was fitted by using the ordinary least squares method (OLS) regression to obtain the model parameters (4)Y=β0+β1X1+β2X2+…+βnXn+ϵRSS=Σ(yi−yi′)2$$\text{Y}={\rm \beta}_{0}+{\rm \beta}_{1}{{\text{X}}_{1}}+{\rm \beta}_{2}{{\text{X}}_{2}}+\ldots +{\rm \beta}_{\text{n}}{{\text{X}}_{\text{n}}}+\epsilon \text{RSS}=\Sigma{{({{\text{y}}_{\text{i}}}-\text{y}_{\text{i}}^\prime})}^{2}$$

Where:(Y) is the dependent varibalbe.(x1,x2,...,xn)are the independent variable.

(β0,β1,β2,βn) is hte regression coefficient,(ϵ)is the regression coefficent.

The core idea of the least squares method is to find a set of regression coefficients that minimize the sum of squared residues between the predicted value and the actual value. The calculation formula of residual sum of squares (RSS) is: (5)RSS=Σ(y1−y′1)2$${\bf RSS}= \Sigma ({\bf y}_1-{\bf y}\prime_1 )^2$$

where: y_i is the actual value and yi′$${\rm y}\prime_{\rm i}$$ is the predicted value.

Second, the Python environment preparation

Before you start, make sure you have the following Python libraries installed: numpy for numerical calculations, pandas for data processing, matplotlib for data visualization, and statsmodels for statistical modeling. Code is as follows:

Figure 6.

Results diagram of linear regression of computing power

From the regression results, we can see the following information:

The degree of fit is R-squared=1.000, indicating an excellent model fit to the data. Adj. R-squared= 0.999, indicating that the adjusted fit is still very high. The overall significance of the model was found to be very high, corresponding to a p-value of 0.0175, indicating that the overall model was statistically significant.

A0=-5.573、λ=0.3477、α=0.1815、β=1.2557,

The p-value of A0 is 0.402, meaning that the intercept term is not statistically significant. The p-value of λ is 0.070, close to the critical value of 0.05, indicating that the effect of time t on ln (Q) is statistically nearly significant. The p-value of α was 0.405, indicating that the effect of capital input on output is not statistically significant. The p-value of β was 0.287, indicating that the effect of labor input on output was not statistically significant.

ln(Q)=-5.573+0.3477t+0.1815ln(K)+1.2557ln(L)

The c-d function expression is:

Q = A0 * e^0.3477t * K^0.1815 * L^1.2557

From the perspective of input and output, λ = 0.3477 indicates that as time goes by, computing power Q grows at an annual rate of about 34.77% per year, which is statistically significant. The positive correlation between A0e ^λ t and Q shows the importance of keeping pace with The Times to the growth of computing power. Although the α and β elastic coefficients are not statistically significant, the explanation from an economic point of view means that when other factors remain unchanged, α =0.1815 indicates that a 1% increase in capital leads to an increase of about 0.1815%.β =1.2557 indicates that a 1% increase in labor leads to about 1.2557% increase in computing power Q.β>α That the number of workers has a greater impact on computing power.

α+β> 1 indicates that there are increasing returns of scale in the production process, and that technological progress makes the production process more efficient over time, resulting in a higher output growth rate. More efficient resource allocation and management can improve productivity and ensure greater returns for invested resources. In the field of digital technology and communications, increasing user and resource input will bring more significant added value. According to the new economic growth theory, the increase in capital and labor input can not only directly increase output, but also may lead to higher productivity growth through technological innovation and knowledge accumulation. The mutual cooperation between capital and labor is crucial. Increasing more capital input can stimulate the enthusiasm of workers and promote the improvement of production efficiency.

Combined with grey correlation analysis, A0e ^λ t> L> K, the progress of technology over time is the main cause of force scale growth, human factors, capital on the force is mainly indirect, reflected in the work enthusiasm, on the progress of technology over time, direct influence is small, comprehensive influence is consistent with grey correlation analysis results. Due to the influence of technology and human resources, the computing power industry can be defined as a knowledge-intensive industry. The introduction of policies to support technological innovation and talent training will improve the output of computing power. It should be particularly noted that the observation volume is only a small sample size of 5 years. In the future, continuous observation and accumulation of more diverse data are needed to improve the accuracy and significance of the statistical data. 2)

Regression analysis on Hypothesis 2

Figure 7.

Schematic diagram of the computing force C-D function

According to Y = AK ^α Q ^β, the log equation ln (Y) = ln (A) + α ln (K) + β ln (Q) was used for regression analysis using python. Judging from the digital industrialization Y1, industrial digitization Y2 analysis graph, R-squared, R-squared, R-squared, F, p-value test results are significant, the model is statistically significant, the fit is good, and the residual values conform to the normal distribution. The dubin test value close to 2 indicates that there is no autocorrelation problem and multiple collinear problem between the data, α is not statistically significant, and the β of computing power input is statistically significant, indicating that the computing power input is the main driving factor of the development of industrial digitalization and digital industrialization, and the capital factor is the indirect factor. The parameters are:

Digital industrialization A1= -11.2813 α 1= -0.199 β 1=1.6

A2= -81.156 α 2= -2.312, β 2=9.63

ln(Y1) = -11.2813 - 0.2 ln(K) + 1.61 ln(Q)

ln(Y2) = -81.156 - 2.312 ln(K) + 9.63 ln(Q)

According to the results of the regression analysis, the elasticity coefficient β of the computing force resource is positive and significant, showing that the computing force input has a significant positive impact on the output. This is consistent with the conclusion in the qualitative analysis, that is, computing power resources play a key role in promoting the development of digital economy. From the numerical comparison of β 2> β 1, the contribution rate of computing power is greater than digital industrialization in promoting the development of industrial digital, indicating that the impact of computing power on industrial digital is greater than that of digital industrialization. Specifically, as long as there is a 1% change in computing resources, the total output value of digital industry will change by 9.63% accordingly, and the total output value of digital industrialization will also change by 1.61%. The formation of data assets and the vertical development of the industry are exactly the significant effect brought by the improvement of computing power. In the fields of artificial intelligence, blockchain and big data, the high elasticity coefficient of computing power resources further verifies its importance in realizing intelligence, data security and data processing. From the scale effect, α 1 + β 1=1.41, α 2 + β 2=7.318 are greater than 1 that digital industrialization, digital industry is increasing scale reward type, although alone, information transmission, software and information technology services investment (K) may have a negative impact on industrial digital output value, but combined with the force (Q), the overall effect is more positive for the force of the digital economy.

By combining C-D function and gray correlation analysis, the connection between digital industrialization, industrial digitalization, capital and computing power can be discussed more comprehensively. Under the consideration of capital, although it will bring a negative impact on output in some cases, its close gray relationship reveals the need to make full use of capital in resource allocation and infrastructure construction. In the context of the booming development of digital economy, although the importance of computing power as a driving force cannot be ignored, the current investment in computing power is still insufficient. In the future, relevant investment should be increased to promote technological upgrading and digital transformation, and achieve the sustained growth of digital economy. Digital industrialization Python code is as follows:

Figure 8.

Results of linear regression calculation for digital industrialization

The industry digital Python code is as follows:

Figure 9.

Results diagram of linear regression calculation of industrial digital log equation

Figure 10.

A 3D rendering of the C-D function of digital industrialization and industrial digitalization