Industrial transfer and regional economy coordination based on multiple regression model

Simple speaking, refers to the enterprise as the main body of industry transformation of the economic activity, belongs to a country or region in order to promote the industry innovation, factors such as the condition of resource supply or demand change, part in the mature industry, innovation, or recession stage of development from one country or area to another country or region of a kind of economic behaviour. The specific mechanism is shown in the figure below:

Fig. 1

Based on the framework of neoclassical ideas, the corresponding mathematical model analysis shows that suppose there are only two countries in the world, namely country A and country B and labour as the only factor of production, there are only two kinds of products, namely product 1 and product 2. Within the country, the products produced by all enterprises have the same production function, and the markets of each country are perfectly competitive. In the case of clear production function F, the production quantity of product I in country J is Y_ij, and the input quantity of labour is L_ij, then we can get [1]: Yij=Fij(Lij)(i=1,2j=a,b) {Y_{ij}} = {F_{ij}}\left( {{L_{ij}}} \right)\left( {i = 1,2j = a,b} \right) Also Fij′(Lij)=Fij(Lij)/Lij F_{ij}^\prime\left( {{L_{ij}}} \right) = {F_{ij}}\left( {{L_{ij}}} \right)/{L_{ij}} Combined with the analysis shown in Figure 2, in the context of long-run equilibrium, the marginal output or marginal revenue of each sector in each country is equal to its average output and average income. L_ij represents the minimum cost and labour input of the product's production in each country, which can be calculated as L_ij0. Consider that the total output of product I produced in country J can be regarded as Z_ij, and the consumption propensity and production function of the two countries are the same, then the Douglas production function can be defined [2].

Fig. 2

In the case of the same propensity to consume products, it is assumed that the income of product I in country J is I_j and the actual demand of product I is X_ij, then we can get: Nij=aij(Lj/Lij0) {N_{ij}} = {a_{ij}}\left( {{L_j}/{L_{ij0}}} \right)

According to the analysis of the above formula, the sum of the number of enterprises in the industry I in country J can use the ratio of the expenditure propensity of the I industry product and the minimum input labour cost of the product I in the total amount of labour input.

According to the analysis of the economic development situation of the north and south of Jiangsu area at present, due to its regional advantages, in the process of industrial innovation and continuous expansion of industrial scale, enterprises in the southern region have gradually gathered. In this way, they can not only obtain more professional products and services but also build a shared labour market. Although the northern region has also achieved certain economic benefits, the overall development level has decreased significantly compared with the southern region. Especially after entering into the 1990s, the economic differences between the north and the south became more and more obvious and gradually formed a ladder development situation from the south to the north. Such unbalanced development has led to serious imbalances in infrastructure, medical investment and education levels in both regions. Nowadays, the land cost and labour force in people's areas are increasing with the economic development, and the overall consumption level is also increasing. Moreover, with the expansion of the industrial scale, development problems such as environmental pollution and resource shortage are gradually introduced. In this context, transferring some industries to the northern region and sharing them with the southern economic development can not only solve the problems of the current industrial economic development but also promote the pace of economic construction and innovation in the northern region [3].

By studying the strength of the economic connection between the two regions, it can be seen that the degree of economic connection between the two regions is directly proportional to the mass of the two regions and inversely proportional to the distance between the centres of the two regions. Consider building a model as shown below: Rt=PnVnPSVS/D2 {R_t} = \sqrt {{P_n}{V_n}} \sqrt {{P_S}{V_S}} /{D^2} Among them, R_t represents the maximum possible strength of the economic connection between the two places. P_n is the total population of northern Jiangsu, in 10,000 people; P_S represents the total population in the south of Jiangsu in 10,000; V_N represents the total industrial output in northern Jiangsu Province, in units of 100 million yuan; V_S represents the total industrial output in the south of Jiangsu Province, and the unit is 100 million yuan.

Combined with the investigation and analysis of the strength of the regional economic connection between the two sides in recent years, we can see that it has been showing a gradual upward trend. This also proves the importance of industrial transfer in economic development and industrial upgrading.

The classical multiple regression model is shown as follows: Y=a0+a1x1+a2x2+......+apxp+ε Y = {a_0} + {a_1}{x_1} + {a_2}{x_2} + ...... + {a_p}{x_p} + \varepsilon It can visually present the predictor Y and factors x₁, x₂... The linear relationship between.

Since there are few cases of a linear relationship between predictors and factors in actual data and most of them are non-linear, in order to expand the application range of the model, factors will be transformed based on several forms of function transformation and then selected by means of stepwise regression. The details are as follows:

First of all, it is necessary to define the function conversion form that can be selected based on the physical background of the data:

Linear function:y = A + Bx

Second, for y and x_i (I = 1,2,3 . . . , p), construct six unary regression models, as shown above, compare the value of F and select the regression model corresponding to the maximum value of F as the function transformation form of x_i, and get: yi=fi(xi),i=1,2,...,p {y_i} = {f_i}\left( {{x_i}} \right),i = 1,2,...,p And finally, we want y to be equal to y₁, y₂, y₃ . . . YP implements stepwise regression and builds x₁, x₂, x₃... The nonlinear multiple regression model of XP.

In this method, only 6p unary regression is needed to select the function transformation form of the factors.

In the derivation of model establishment, the southern and northern parts of Jiangsu were taken as the whole regional economy for in-depth discussion and the production function equation as shown below can be obtained, respectively: St=S(LSt,KSt)Bt=B(LBt,KBt,St*)Yt=St+Bt \matrix{{{S_t} = S\left( {{L_{{S_t}}},{K_{{S_t}}}} \right)} \hfill \cr {{B_t} = B\left( {{L_{{B_t}}},{K_{{B_t}}},S_t^*} \right)} \hfill \cr {{Y_t} = {S_t} + {B_t}} \hfill \cr }

In the above formula, S_t represents the total production value of the southern region in the period of t, while B_t represents the total production value of the northern region in the period of t. L_St represents the size of the labour force in the southern region at the time t and L_Bt represents the size of the labour force in the northern region at the time of t; K_St represents the investment level of fixed assets in the southern region in the period t; K_Bt represents the investment level of fixed assets in the northern region in the period t; represents the total industrial production value in the southern region in the period t.

At the same time, since the externalities of industrial transfer in both regions can only be reflected after a period of time, the formula can be obtained as follows:

In this paper, the adaptive expectation model is used to optimise the above contents. Assuming that there is no relationship between the expected external influencing factor and S_t at the present stage, it will be limited by the expected influencing factor at the previous stage, and it can be obtained as follows: St*=θSt+(1−θ)St−1* S_t^* = \theta {S_t} + \left( {1 - \theta } \right)S_{t - 1}^*

In the above formula, 0 < θ ≤ 1, it is proved that the externality of industrial transfer in the southern region meets the following conditions: First, the externality of total output quantity S_t to the northern region is 1, and meets the condition of θ∑i=0∞(1−θ)i=1 \theta \sum\limits_{i = 0}^\infty {\left( {1 - \theta } \right)^i} = 1 . In this case, the essence of the adaptive expectation model is to assume that the externality of industrial transfer from the south to the north should conform to the geometric lag condition. Second, the externality coefficient a_j of industrial transfer will decrease with time. Combined with the above formula analysis, it can be seen that the externality coefficient of the total output quantity of the southern region in the lag stage will show a geometric decreasing trend according to the ratio of (1−θ).

Thus, the production function of the two regions is substituted into the formula to obtain the total differential, and the following can be obtained: dYt=SLdLSt+SKdKSt+BLdLBt+BKdKBt+BS*dS* d{Y_t} = {S_L}d{L_{{S_t}}} + {S_K}d{K_{{S_t}}} + {B_L}d{L_{{B_t}}} + {B_K}d{K_{{B_t}}} + {B_{{S^*}}}d{S^*}

At this point, it is assumed that the marginal elements of the two regions are different, so it can be concluded that: SLBL=SKBK=1+δ {{{S_L}} \over {{B_L}}} = {{{S_K}} \over {{B_K}}} = 1 + \delta

Substituting the above two formulas into the analysis, we can get: dYt=(1+δ)BLdLSt+(1+δ)BKdKSt+BLdLBt+BKdKBt+BS*dS*=BLdL+BKdK+BS*dS*+δ(BLdLSt+BKdKSt)=BLdLt+BKdKt+BSt*dSt*+δ1+δdSt \matrix{{d{Y_t} = \left( {1 + \delta } \right){B_L}d{L_{{S_t}}} + \left( {1 + \delta } \right){B_K}d{K_{{S_t}}} + {B_L}d{L_{{B_t}}} + {B_K}d{K_{{B_t}}} + {B_{{S^*}}}d{S^*}} \hfill \cr { = {B_L}dL + {B_K}{d_K} + {B_{{S^*}}}d{S^*} + \delta \left( {{B_L}d{L_{{S_t}}} + {B_K}d{K_{{S_t}}}} \right)} \hfill \cr { = {B_L}d{L_t} + {B_K}d{K_t} + {B_{S_t^*}}dS_t^* + {\delta \over {1 + \delta }}d{S_t}} \hfill \cr } At this point, let dSt*=θμtdSt dS_t^* = \theta {\mu _t}d{S_t}

In the above formula, θ represents the multiplier of the external impact factor of industrial transfer, and μ_t represents the proportion of the total GDP of the industrial sector in the local GDP. Continuing into the above formula, we can get: dYt=BLdLt+BKdKt+BSt*θμtdSt+δ1+δdSt=BLdLt+BKdKt+(BSt*θμt+δ1+δ)dSt \matrix{{d{Y_t} = {B_L}d{L_t} + {B_K}d{K_{{t_{}}}} + {B_{{S_t}^*}}\theta {\mu _t}d{S_t} + {\delta \over {1 + \delta }}d{S_t}} \hfill \cr { = {B_L}d{L_t} + {B_K}d{K_t} + ({B_{S_t^*}}\theta {\mu _t} + {\delta \over {1 + \delta }})d{S_t}} \hfill \cr }

If there is a linear relationship between the marginal production efficiency of labour and the per capita production within the scope of the overall economy, then we can get: BL=β⋅(Y/LB) {B_L} = \beta \cdot \left( {Y/{L_B}} \right)

Substituting it into the above formula and dividing both sides of the equation by Y_t, and you get: dYtYt=BLdLtYt+BKdKtYt+(BSt*θμt+δ1+δ)dStSt⋅StYt {{d{Y_t}} \over {{Y_t}}} = {{{B_L}d{L_t}} \over {{Y_t}}} + {{{B_K}d{K_t}} \over {{Y_t}}} + \left( {{B_{S_t^*}}\theta {\mu _t} + {\delta \over {1 + \delta }}} \right){{d{S_t}} \over {{S_t}}} \cdot {{{S_t}} \over {{Y_t}}}

Combined with the production function formula analysis of the southern region, the influence of the externality of local output transfer on the total production quantity in the northern region conforms to the following conditions: Bt=B(LBt,KBt,St*)=(St*)ωφ(LBt,KBt) {B_t} = B\left( {{L_{{B_t}}},{K_{{B_t}}},S_t^*} \right) = {\left( {S_t^*} \right)^\omega }\varphi \left( {{L_{{B_t}}},{K_{{B_t}}}} \right)

By continuing to deduce and analyse the above formula, the marginal production efficiency of the external influencing factor can be determined. The specific formula is as follows:

Putting the above formula into the linear relationship between the marginal productivity of labour and the scope of the overall economy, we can get: dYtYt=BLdLtYt+BKdKtYt+δ1+δ⋅dStSt⋅StYt+ωθμtBtSt*⋅dStSt⋅StYt {{d{Y_t}} \over {{Y_t}}} = BL{{d{L_t}} \over {{Y_t}}} + {B_K}{{d{K_t}} \over {{Y_t}}} + {\delta \over {1 + \delta }} \cdot {{d{S_t}} \over {{S_t}}} \cdot {{{S_t}} \over {{Y_t}}} + \omega \theta {\mu _t}{{{B_t}} \over {S_t^*}} \cdot {{d{S_t}} \over {{S_t}}} \cdot {{{S_t}} \over {{Y_t}}}

Further recursive iteration processing is carried out for the external impact factor, and the following can be obtained:

The above formula satisfies the condition of θ∑i=0∞(1−θ)i=1 \theta \sum\limits_{i = 0}^\infty {\left( {1 - \theta } \right)^i} = 1 , so in essence, can be regarded as the weighted average value of the total production quantity of the southern region at the present and previous stages. Combined with the above formula analysis, it is further verified that there is an equivalent relationship between the adaptive expectation process and the Loyck geometric lag model. Thus, the above formula can be transformed into:

It represents the basic dynamic externality model studied in this paper.

According to the analysis of the basic dynamic externality model, multiplying both sides of it by Y_t, we can get: Yt=2θBt+θδ1+δSi+ωθμtBtLnSt+(1−θ)YtLnYt−1 {Y_t} = 2\theta {B_t} + \theta {\delta \over {1 + \delta }}{S_i} + \omega \theta {\mu _t}{B_t}Ln{S_t} + \left( {1 - \theta } \right){Y_t}Ln{Y_{t - 1}}

In the above formula, θ analyses the distribution of the externality of the industrial output value of the southern region in the lag region, W analyses the externality elasticity of the industrial output value of the southern region on the northern region and σ evaluates the difference change of the editorial factor productivity between the southern region and the northern region. Combined with the analysis of the derivation results of the above models, an empirical model is built for the spillover effect of technology externalities in the southern region as shown below: Yt=a1Bt+a2St+a3BtLnSt+a4YtLnYt−1+ξt {Y_t} = {a_1}{B_t} + {a_2}{S_t} + {a_3}{B_t}Ln{S_t} + {a_4}{Y_t}Ln{Y_{t - 1}} + {\xi _t}