Synergy analysis of agricultural economic cycle fluctuation based on ant colony algorithm

2.1 Transmission mechanism of economic cycle synergy trade

The coordinated fluctuation of the agricultural economic cycle is a phenomenon of economic cycle fluctuation. The theoretical explanation of this phenomenon should return to the theory explanation of economic cycle fluctuation mechanism. It is believed that the coordinated fluctuation of Chinese agricultural economic cycle is the internal transmission mechanism of the agricultural economy and the conduction to the economy. Cyclic fluctuations produce the result of an external impact mechanism. Exploring the transmission law of the international economic cycle and agricultural economic fluctuations is of great significance for further analysis of the coordinated fluctuations for the agricultural economic cycle [6].

2.2 Intermediate product production

In each country, n = H, F, a continuous monopolistic competitor employs all of the labor supply L_ne in the country to produce intermediate goods j, where N_nt represents the set of intermediate products available. Total Factor Labor Productivity (TFP) is described as Z_nt, indicating the efficiency of labor per unit of domestic labor, following the AR(I) process:

where, ρ_n ∈ (0, 1).

The heterogeneity of the agricultural industry is reflected in the differences in production techniques expressed in terms of relative productivity z. A domestic agricultural enterprise with a relative productivity of z employs 1 unit of labor to produce Z_ntz units of output. The unit cost of production is given as ω_nt /Z_ntz, where ω_nt = W_nt /P_nt is the relative real wage,W_nt and P_nt are the nominal wage and price indices of country n [7].

Intermediate product manufacturers can serve both domestic and foreign markets. Two types of export costs are introduced: the transportation cost of agricultural products τ_t ≥ 1, which mainly affects the margin of trade depth, and the fixed entry cost f_X,t measured by effective labor units mainly affects the margin of trade. The fixed cost expressed in kind is paid separately in each period, and the fixed cost of export as a trade barrier mainly affects the margin of trade.

In countries of n = H, F, the demand function of the final producer faced by all intermediate production enterprises is given, so a price is added to the reaction marginal cost. The transportation costs that the products need to pay for shipment to foreign markets τ_t divide the domestic and international markets, and the prices of the two countries are different. pn,tD and pn,tD represent the domestic sales price and export price of the agricultural products of n countries, respectively. The true price relative to the target market nominal price index is expressed as:

where Q_t is the real exchange rate and θθ−1is the constant addition.

Due to the fixed export costs, enterprises with low productivity z may decide not to export at any time. The enterprise breaks down its total profit π_nt into domestic market sales profit πntD and potential foreign export profits. The total profit of agricultural products in the period t is:

In each period, n countries have NntD domestic enterprises for production. Among these enterprises, NntX=1−G(zn,tX)NntD enterprises export agricultural products to foreign markets. As long as the productivity level of the enterprise is higher than the critical productivity level zntX=infz:πntX(z)>0, these enterprises will continue to export.

2.3 Final product production

In countries of n = H, F, a fully competitive enterprise uses capital K_nt and a collection of tradable intermediate products at home and abroad, which is the material capital Mntα, and generating a non-tradeable final product

Intermediate products are combined according to the CES production function to generate physical capital:

where θ > 1 is the symmetry alternative elasticity between the intermediates. The CES production function introduces diversity preferences: when expenditure y_njt remains constant, the larger the number N_nt of species employed, corresponds to an increase in productivity levels.

The final product manufacturer chooses y_njt to maximize profits:

where P_nt is the final product price index and is expressed as:

By merging the final product producer’s demand for each intermediate y_njt, then:

2.4 Family department

In countries of n = H, F, a representative household sector into:

The productivity assumption assumes that the lower limit is the z_min Pareto distribution and the shape parameter is k > θ − 1. Based on the above assumptions, the average production productivity threshold for domestic production and exports is obtained:

where, v=kk−(θ−1)1θ−1.

The number consumes end products, provides labor and leases capital to intermediate producers or for savings. The family department maximizes the expected utility function of a lifetime:

The budget constraints faced are:

where C_nt is consumption, β is subjective discount factor, ψ is labor supply elasticity, P_nt is price index, I_nt is nominal investment, ω_nt is wage, Π^T_nt is the total profit of all enterprises in n countries, B_nt is family sector in period of t−1 to provide the total amount of loans that can be repaid in the period t, R^k_nt is the risk-free probability, rr is the rent price of capital, and K_nt is the capital supply obtained as follows:

where δ is the capital depreciation rate. The decision-making problem in the household sector is to choose consumption, labor attacks, and capital and maximize the utility of formula (14) under the constraints of formula (15).

There are an unlimited number of potential entrants in each period. The entrant is forward-looking and maximizes his profit πnt(z)=πntD(z)+πntX(z). All profits are expressed in terms of the number of final products under actual conditions.

If the enterprise z exports, there are:

Set average productivity levels and generalize all the information about the productivity distribution associated with macroeconomic variables, then:

The expected corporate value after the entry of the entrant is described by the discounted value of the expected profit stream:

New entrants enter the market until the average corporate value equals the entry cost. The free entry conditions are:

Assume that the entrant during the period of t begins production in the period of t + 1. The number of domestically produced varieties is:

The budget constraint is transformed of export intermediates is:

It is worth noting that when other conditions remain unchanged, as the number of export varieties increases, the more domestic manufacturers enter, the lower the average productivity threshold of manufacturers’ exports.

Assuming that the financial sector is closed, trade is balanced in all phases, and the total exports of n countries are equal to the total imports, that is, the balance of trade is balanced:

The final product of each country is either used for consumption or for investment, and the market clearing conditions are:

The demand for domestic and foreign varieties is equal to its supply:

The balance of foreign intermediates is similar to the domestic situation. For countries of n = H, F, the general equilibrium of the symmetry of the economy is set as: exogenous random sequence {Z_nt}, initial vector {Zn0,Nn0D,Kn0}, a set of different parameters {p_n} for two countries, average price and wage sequence {Qt,Pnt,Rnt,ωnt}t=0∞, a set of intermediate price p~ntD,p~ntXt=0∞, one group total sequence{Y_nt , ỹ_nt , I_nt}, intermediate product yield sequence {Y_nt , ỹ_nt , I_nt}, intermediate product quantity sequence {yntD,yntX}t=0∞, domestic production and export average productivity threshold {zn,tX,zn,tD}t=0∞, profit and enterprise value sequence {Πnt,d~nt,v~nt}t=0∞, capital accumulation rule {NntD,NntX,Zn,t+1,Knt}t=0∞.

It can be seen from formula (4) and formula (5) that production efficiency and total factor productivity (TFP) depend on the quantity of domestic and foreign intermediate products used to produce the final product. By averaging the firm variable (19), it can be found that the TFP in the model consists of two parts: the total productivity impact determines the amount of the exogenous part and the intermediate product used to produce the final product and the average productivity of each intermediate product.

Based on the above process, the transmission mechanism of the coordination of agricultural economic cycle fluctuations is given:

A positive productivity shock in the home country creates a demand-supply spillover effect. Through this effect, the demand for foreign intermediate goods by domestic final product producers will increase, and the output of foreign economies will increase. This is also a channel of communication presented in the traditional economic cycle model.

2.5 Research on coordination characteristics of economic cycle based on prosperity index

Through the fluctuation range of the economy, the characteristics of the industry cycle fluctuations are studied, and the composite index method that can specifically describe the fluctuation range of the economic climate of the industry which is used to calculate the industry sentiment index.

To find the symmetrical rate of change of the indicator and standardize it.

Let indicator C_i(t) as the i-th indicator and i = 1, 2, · · · , K as the indicator number. The ratio sequence is chosen, and the rate of change C_i(t) is taken as the Y_i(t) first order difference:

In order to prevent the index of abnormality of the rate of change from gaining a dominant position in the composite index, it is necessary to standardize the rate of symmetrical change of each index so that the average absolute value is 1. The normalization factors are given below:

Standardizing C_i(t) with Ai to obtain a standardized rate of change Si(t):

The average rate of change of the consistent indicator group R(t):

To calculate the initial consistent composite index, let, then agree on the composite index:

Only the consensus synthesis index is required, and there is no need to adjust the trend for the leading and lagging indicator groups to obtain the composite index with the base year of 100:

where I¯is the average of the base year and 2018 is the base year.

The cumulative annual growth rate of agricultural added value, the cumulative year-on-year growth rate of main business income, and the cumulative year-on-year growth rate of fixed asset investment were the prosperous indicators, and the consensus synthesis indexes of agricultural and agricultural sub-sectors were established [8].

2.6 Descriptive statistical analysis of industry consensus synthetic index

In order to further empirically test the synergy of Chinese agricultural economic cycle fluctuations, study the law of Chinese agricultural economic cycle fluctuations, and analyze the statistics of economic climate cycle fluctuations by describing the statistics of agricultural economic sentiment consistent synthetic index trend fluctuation time series. The wave component and trend component are separated to eliminate the impact of common trends among various industries on the analysis of fluctuation correlation, and to better discover the law of economic fluctuation. The HP filtering method, which is widely used in the world to decompose the economic time series trend, is selected to decompose the trend fluctuations of the economic time series of various industries to obtain the wave components, and perform spearman correlation analysis on the wave components to verify the fluctuation of various industries.

Based on modern economic growth theory, the economic cycle of an economy can be seen as consisting of long-term trend components and short-term fluctuations. Before conducting the study of the volatility characteristics of the economic cycle, it is necessary to determine the separation of the long-term trend components to obtain the actual economic cycle fluctuation components.

Let the economic time series Y = {y₁, y₂, · · · , y_n}, the long-term trend term is Yt={y1t,y2t,⋯,ynt},and is the sample length of the time series. The “smooth” feature of long-term trend components allows the extraction of trend component problems from time series to be defined as the solution variance minimum:

where λ > 0 is the smoothing parameter, indicating the weight of the decomposition trend component, and c(L) is the lag operator polynomial, the expression is:

Bring formula (35) into (34), the minimum function is:

where ∑i=1n(yi−yit)2 is a measure of sequence fluctuations and λ∑i=1n(yit−yi−1t)−(yit−yi−2t)2} is a measure of longterm trend smoothness. Here you need to choose the appropriate λ value, different λ values determine the different period fluctuations and smoothness, which has a direct impact on the accuracy of the results. The value of λ = 100 usually takes λ = 1600 when the time series is annual data; for monthly data, takeλ = 15200.

2.7 Analysis of the synergistic characteristics of economic cycle fluctuations

Before the analysis of the synergistic characteristics of the economic cycle of the agricultural industry, the statistical analysis of the separated wave components is carried out, and the Spearman correlation coefficient between the sequences is selected as the correlation index. The Spearman method was chosen because it provides a better measure of the correlation between non-normally distributed time series and nonlinear relationships. The Spearman correlation coefficient is calculated as:

where R_i and Q_i are the ranks of two random variables x_i and y_i in their corresponding vectors (x₁, · · · , x_n) and (y₁, · · · , y_n), respectively.

The Spearman correlation analysis method has some limitations. The method requires the variable to conform to a normal distribution, and is not credible if the condition is not satisfied. To solve this problem, the ant colony algorithm [9] is introduced, and the solution of the wave correlation coefficient can be regarded as the optimization process of the following function:

This is a constrained minimum problem whose domain is a contiguous domain, namely:

The objective function is:

The restrictions are:

The goal is to find a set of optimal values within the normal range of the fluctuation correlation coefficient so that the minimum value is obtained under the target value sub-constraint.

Randomly virtualized Ṅ individual workers in the feasible solution range (take Ṅ k̇ values randomly within the range of k̇, and set the evaluation function value to be the corresponding target value f_o. The transfer probability of each ant (ie from the definition of the probability that the space of the ant ξ is transferred to the space where the ant ξ is located is different from other combinations of the optimized path class problem, and is set as follows:

among which, the track strength τ_ζξ is converted to τ_ζ , which represents the pheromone size of the ant ζ , and can also be referred to as the field attraction intensity of the ant ζ . The heuristic pheromone η_ξζ = f_ξ − f_ζ represents the difference value of the objective function. The parameters α and β are used to describe the relative influence of pheromones and heuristic pheromones, respectively.

When the ants in the population have constructed the path, the pheromone on each side will be updated. All pheromones reduce the size of a constant factor or the retention strength of a pheromone. Add pheromones to the side where the ants pass. The evaporation of pheromones or the retention of pheromones is performed according to the following formula:

where ᑮ is called the vegetative evaporation rate and 0 < ᑮ≤ 1. The role of the parameter ᑮ is to avoid the infinite accumulation of pheromones. If an edge is not selected by any other ant, the pheromone on the edge will be decremented exponentially. After the pheromone evaporation step, all ant colonies release pheromones on the edges they pass, and the pheromone intensity update formula is:

2.8 Quantitative calculation model for cooperative estimation

The Fisher-z conversion is performed on the optimal Spearman correlation coefficient. The transformed results represent the degree of economic cycle synergy, and consider the geographical distance, economic space spillover, fiscal policy synergy, regional income gap and geographical neighbors to construct a quantitative calculation model for agricultural economic cycle synergy evaluation.

The economic cycle synergy degree Z_ϕφ is based on the Spearman correlation coefficient. The specific transformation formula is as follows:

among which, the correlation coefficient ρ_ϕφ is calculated from the periodic component of the actual output; the degree of coincidence I_ϕφ is expressed according to the proportion of the economic period between the regions in the same period.

The geographical factor D_ϕφ represents the geographical distance between the ϕ region and the φ region. The geographical distance affects the flow cost of agricultural products and factors, changes the trade flow between regions, and affects the degree of synergy in the economic cycle. Introducing geographical distance factors into the model, it is generally believed that the further the geographical distance between regions, the higher the trade cost required, and the lower the economic cycle synergy.

The economic scale S_ϕφ will also cause changes in trade flows between regions, which in turn will affect the degree of synergy in the economic cycle [10]. The size of the economy should be introduced into the model. It is generally believed that the larger the size of the inter-regional economy, the greater the trade flows generated, and it is expected to increase the synergy of the economic cycle. The economic scale is the sum of the regional GDP between the two provinces. The specific calculation formula is as follows:

where S_ϕφ represents the economic size between the region ϕ and the region φ, GDP_ϕt and GDP_φt are used to describe the regional GDP of the ϕ region and the region, respectively.

The fiscal policy coordination degree is c_ϕφ. To a certain extent, the formulation of national macroeconomic policies can “smooth” the fluctuations of the economic cycle and become the main factor for the coordinated impact of the inter-regional economic cycle. Therefore, the coordination degree of fiscal policy in macroeconomic policy is introduced into the model as an explanatory variable. Experts predict that the higher the inter-regional fiscal policy coordination, the more similar the economic cycle [11]. The coordination of financial and economic policies is represented by the gap between regional fiscal expenditures, and the calculation formula is as follows:

among which, c_ϕφ is used to describe the fiscal policy coordination degree between region ϕ and region φ, and f_ϕt and f_φt respectively represent the proportion of provincial finances to the GDP of other regions.

The inter-regional income gap also affects trade flows between regions and thus the degree of economic cycle synergy [12], which is introduced into the model. It is expected that the widening income gap between regions will reduce the degree of synergy in the economic cycle. The income gap is set as the sum of the per capita disposable income between the two provinces divided by the per capita disposable income. The specific formula is:

where Y_ϕφ represents the income gap between the regional ϕ and φ regions, and y_ϕt and y_φt represent the per capita disposable income of the ϕ and φ regions, respectively.

The specialization of production will increase regional differences [13], affecting the synchronicity of the regional economic cycle, and the same factor is added to the model. The difference in industrial structure is constructed according to three industrial differences between provinces. The calculation formula is as follows:

where Sis_ϕφ represents the industrial structure difference between the region and the region φ, and S^k_ϕt and S^k_φt respectively represent the proportion of the industry in the ϕ and φ regions, and k = 1, 2, and 3 represent the primary industry, the secondary industry, and Tertiary Industry.

Geographical proximity B_ϕφ, because the domestic goods and elements can flow freely, the geographically adjacent areas trade relatively close, naturally generate different trade flows, and thus affect the degree of economic cycle synchronization. Set the geographic proximity factor to a dummy variable, the neighboring provinces take a value of 1, and the non-adjacent provinces take a value of 0.

In order to further determine the impact of different regions on the degree of synergy of the agricultural economic cycle [14], introduces dummy variables in the eastern, central and western regions. If the two regions are in the same region, the value is 1, otherwise it is 0.

Considering geographical distance, bilateral economic scale, fiscal policy synergy, regional income gap, geographical neighbors, economic space spillover, government expenditure cycle synergy, investment cycle synergy, international trade cycle synergy and other factors to construct quantitative calculation of agricultural economic cycle synergy estimation model, then:

where Z_ϕφ is the explanatory variable, indicating the degree of synergy in the agricultural economic cycle, D_ϕφ is the geographical distance variable, S_ϕφ is the economic scale variable, c_ϕφ is the fiscal coordination degree variable, Y_ϕφ is the inter-regional income gap variable, Sis_ϕφ is the industrial structure difference variable, B_ϕφ represents the geographical neighbor variable, and Area represents the regional dummy variable [15, 16, 17, 18, 19, 20, 21].