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

$$\begin{array}{rl}{P}_{nt}{C}_{nt}+{I}_{nt}& ={W}_{nt}{L}_{nt}+{\mathrm{\Pi}}_{nt}^{T}+{R}_{nt}^{k}{K}_{nt}+{R}_{nt}{B}_{nt}\\ & -{B}_{n,j+1}\end{array}$$(11)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:

$${\stackrel{~}{z}}_{n}^{D}=v{z}_{min}$$(12)$${\stackrel{~}{z}}_{n}^{X}=v{z}_{nt}^{X}$$(13)where, $v={\left[\frac{k}{k-(\theta -1)}\right]}^{\frac{1}{\theta -1}}.$

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:

$${U}_{t}={E}_{t}\sum _{s=t}^{\mathrm{\infty}}{\beta}^{s}\left(\mathrm{log}({C}_{nt})-\frac{{L}_{ns}^{\psi +1}}{\psi +1}\right)$$(14)The budget constraints faced are:

$$\begin{array}{rl}{P}_{nt}{C}_{nt}+{I}_{nt}& ={W}_{nt}{L}_{nt}+{\mathrm{\Pi}}_{nt}^{T}+{R}_{nt}^{k}{K}_{nt}+{R}_{nt}{B}_{nt}\\ & -{B}_{n,t+1}+{\omega}_{nt}\end{array}$$(15)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:

$${K}_{nt}=(1-\delta ){K}_{n,t-1}+{I}_{nt}/{P}_{nt}$$(16)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 ${\pi}_{nt}(z)={\pi}_{nt}^{D}(z)+{\pi}_{nt}^{X}(z).$ All profits are expressed in terms of the number of final products under actual conditions.

$${\pi}_{nt}^{D}(z)=\frac{{\alpha}^{\theta}}{\theta}{\left({p}_{nt}^{D}(z)\right)}^{1-\theta}{\left(\frac{{Y}_{nt}}{{M}_{nt}}\right)}^{\theta}{M}_{nt}$$(17)If the enterprise *z* exports, there are:

$$\begin{array}{r}{\pi}_{nt}^{X}(z)=\frac{{\alpha}^{\theta}{Q}_{t}}{\theta}{\left({p}_{nt}^{X}(z)\right)}^{1-\theta}{\left(\frac{{Y}_{nt}}{{M}_{nt}}\right)}^{\theta}{M}_{nt}-{\omega}_{nt}{f}_{Xt}/{Z}_{nt}\end{array}$$(18)Set average productivity levels and generalize all the information about the productivity distribution associated with macroeconomic variables, then:

$${\stackrel{~}{z}}_{D}={\left[{\int}_{-min}^{\mathrm{\infty}}{z}^{\theta -1}dG(z)\right]}^{1/(\theta -1)}$$(19)The expected corporate value after the entry of the entrant is described by the discounted value of the expected profit stream:

$${\stackrel{~}{v}}_{nt}={E}_{t}\sum _{s=t+1}^{\mathrm{\infty}}{\left[\beta (1-\delta )\right]}^{s-t}{d}_{ns}^{\stackrel{~}{}}$$(20)New entrants enter the market until the average corporate value equals the entry cost. The free entry conditions are:

$$\begin{array}{r}{\stackrel{~}{v}}_{nt}={\omega}_{nt}{f}_{E,t}/{Z}_{nt}.\end{array}$$(21)Assume that the entrant during the period of *t* begins production in the period of *t* + 1. The number of domestically produced varieties is:

$${N}_{nt}^{D}=(1-\delta )({N}_{n,t-1}^{d}+{N}_{n,t-1}^{E})$$(22)The budget constraint is transformed of export intermediates is:

$$\frac{{N}_{nt}^{X}}{{N}_{nt}^{D}}={z}_{min}^{k}{\left({\stackrel{~}{z}}_{nt}^{X}\right)}^{-k}{\left(\frac{k}{k-(\theta -1)}\right)}^{\frac{k}{\theta -1}}$$(23)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:

$$\begin{array}{rl}& {Q}_{t}{N}_{H,t}^{-X}{\left({\stackrel{~}{\rho}}_{H,t}^{-X}\right)}^{1-\theta}{\left(\frac{{Y}_{Ft}}{{M}_{Ft}}\right)}^{\theta}{M}_{Ft}\\ & ={N}_{H,t}^{X}{\left({\stackrel{~}{\rho}}_{Ft}^{X}\right)}^{1-\theta}{\left(\frac{{Y}_{Ht}}{{M}_{Ht}}\right)}^{\theta}{M}_{Ht}\end{array}$$(24)The final product of each country is either used for consumption or for investment, and the market clearing conditions are:

$${Y}_{nt}={C}_{nt}+{I}_{nt}$$(25)The demand for domestic and foreign varieties is equal to its supply:

$${\stackrel{~}{y}}_{Ht}={y}_{Ht}^{\stackrel{~}{D}}+{y}_{Ht}^{\stackrel{~}{X}}$$(26)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 $\{{Z}_{n0},{N}_{n0}^{D},{K}_{n0}\},$ a set of different parameters *{p*_{n}} for two countries, average price and wage sequence $\{{Q}_{t},{P}_{nt},{R}_{nt},{\omega}_{nt}{\}}_{t=0}^{\mathrm{\infty}},$ a set of intermediate price ${\left\{{\stackrel{~}{p}}_{nt}^{D},{\stackrel{~}{p}}_{nt}^{X}\right\}}_{t=0}^{\mathrm{\infty}},$ one group total sequence*{Y*_{nt} , *ỹ*_{nt} , *I*_{nt}}, intermediate product yield sequence *{Y*_{nt} , *ỹ*_{nt} , *I*_{nt}}, intermediate product quantity sequence $\{{y}_{nt}^{D},{y}_{nt}^{X}{\}}_{t=0}^{\mathrm{\infty}},$ domestic production and export average productivity threshold $\{{z}_{n,t}^{X},{z}_{n,t}^{D}{\}}_{t=0}^{\mathrm{\infty}},$ profit and enterprise value sequence $\{{\mathrm{\Pi}}_{nt},{\stackrel{~}{d}}_{nt},{\stackrel{~}{v}}_{nt}{\}}_{t=0}^{\mathrm{\infty}},$ capital accumulation rule $\{{N}_{nt}^{D},{N}_{nt}^{X},{Z}_{n,t+1},{K}_{nt}{\}}_{t=0}^{\mathrm{\infty}}.$

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:

$$\begin{array}{rl}& TF{P}_{nt}\\ & ={z}_{nt}\left\{\frac{1}{{N}_{nt}^{D}+{N}_{nt}^{X}}\left\{{N}_{nt}^{D}({\stackrel{~}{z}}_{nt}^{D}{)}^{\theta -1}+{N}_{n,t}^{X}({Z}_{n,t}/{z}_{n,t}^{X}\tau /Q))\right\}\right\}\end{array}$$(27)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.

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