# 粮食最低收购价的预测Forecast of Minimum Purchase Price of Grain

DOI: 10.12677/SA.2018.72018, PDF, HTML, XML, 下载: 711  浏览: 1,954

Abstract: In order to solve the food problem, during “China’s 12th Five Year Plan” the National Develop-ment and Reform Commission issued the minimum purchase price of grain, “China’s 13th Five Year Plan” to continue to implement and improve this system. In this paper, the relationship between planting area and grain minimum purchase price is analyzed by linear regression, gray relational analysis, modified index curve method, logistic model. We find out the law of grain price change and reasonable range, and put forward the optimal decision-making and sugges-tion of regulating grain planting area.

1. 引言

2. 理论模型的选取

2.1. 灰色关联度模型

${x}_{0}=\left\{{x}_{0}\left(t\right),t=1,2,\cdots ,n\right\}=\left({x}_{0}\left(1\right),{x}_{0}\left(2\right),\cdots ,{x}_{0}\left(n\right)\right)$

${x}_{i}=\left\{{x}_{i}\left(t\right),t=1,2,\cdots ,n\right\}=\left({x}_{i}\left(1\right),{x}_{i}\left(2\right),\cdots ,{x}_{i}\left(n\right)\right),\text{\hspace{0.17em}}i=1,2,\cdots ,m$

${\xi }_{i}\left(k\right)=\frac{\underset{s}{\mathrm{min}}\underset{t}{\mathrm{min}}|{x}_{0}\left(t\right)-{x}_{s}\left(t\right)|+\underset{s}{\rho \mathrm{max}}\underset{t}{\mathrm{max}}|{x}_{0}\left(t\right)-{x}_{s}\left(t\right)|}{|{x}_{0}\left(k\right)-{x}_{s}\left(k\right)|+\underset{s}{\rho \mathrm{max}}\underset{t}{\mathrm{max}}|{x}_{0}\left(t\right)-{x}_{s}\left(t\right)|}$

${r}_{i}=\frac{1}{n}\underset{k=1}{\overset{n}{\sum }}{\xi }_{i}\left(k\right)$ (1)

2.2. 回归模型

$y={\beta }_{0}+{\beta }_{1}{x}_{1}+\cdots +{\beta }_{m}{x}_{m}$ (2)

$y={\beta }_{0}+{\beta }_{1}\mathrm{ln}{x}_{1}+{\beta }_{2}\mathrm{ln}{x}_{2}+\cdots +{\beta }_{m}\mathrm{ln}{x}_{m}+\epsilon$

2.3. 预测模型

$\stackrel{^}{y}=K+a{b}^{t}$ (3)

1) 初期增长速，随后增长率逐渐降低。

2) 当 $K>0,a<0,0 时， $t\to \infty ,a{b}^{t}\to 0$ ，即 ${\stackrel{^}{y}}_{t}\to K$

${s}_{1}=\underset{i=1}{\overset{n}{\sum }}{y}_{i},{s}_{2}=\underset{i=n+1}{\overset{2n}{\sum }}{y}_{i},{s}_{3}=\underset{i=2n+1}{\overset{3n}{\sum }}{y}_{i}$

$\left\{\begin{array}{l}{s}_{1}=\underset{i=1}{\overset{n}{\sum }}\left(K+a{b}^{t}\right)=mK+ab\left(1+b+{b}^{2}+\cdots +{b}^{n-1}\right)\\ {s}_{2}=\underset{i=n+1}{\overset{2n}{\sum }}\left(K+a{b}^{t}\right)=mK+a{b}^{n+1}\left(1+b+{b}^{2}+\cdots +{b}^{n-1}\right)\\ {s}_{3}=\underset{i=2n+1}{\overset{3n}{\sum }}\left(K+a{b}^{t}\right)=mK+a{b}^{2n+1}\left(1+b+{b}^{2}+\cdots +{b}^{n-1}\right)\end{array}$

$\left(1+b+{b}^{2}+\cdots +{b}^{n-1}\right)\left(b-1\right)={b}^{n}-1$

$\left\{\begin{array}{l}{s}_{1}=mK+ab\frac{{b}^{n}-1}{b-1}\\ {s}_{2}=mK+a{b}^{n+1}\frac{{b}^{n}-1}{b-1}\\ {s}_{3}=mK+a{b}^{2n+1}\frac{{b}^{n}-1}{b-1}\end{array}$

$\left\{\begin{array}{l}b={\left(\frac{{s}_{3}-{s}_{2}}{{s}_{2}-{s}_{1}}\right)}^{1/n}\\ a=\left({s}_{2}-{s}_{1}\right)\frac{b-1}{b{\left({b}^{n}-1\right)}^{2}}\\ K=\frac{1}{m}\left[{s}_{1}-\frac{ab\left({b}^{n}-1\right)}{b-1}\right]\end{array}$

$\frac{{y}_{t+1}-{y}_{t}}{{y}_{t}-{y}_{t-1}}\approx b$

2.4. Logistic模型

Logistic曲线的一般数学模型是

$\frac{\text{d}y}{\text{d}t}=ry\left(1-\frac{y}{L}\right)$

$y=\frac{L}{1+c{\text{e}}^{-rt}}$

${y}_{t}=\frac{1}{K+a{b}^{t}},\text{\hspace{0.17em}}K>0,\text{\hspace{0.17em}}a>0,\text{\hspace{0.17em}}0 (4)

$\frac{1/{y}_{t+1}-1/{y}_{t}}{1/{y}_{t}-1/{y}_{t-1}}\approx b$

Logistic曲线中参数估计方法如下：

${{y}^{\prime }}_{t}=\frac{1}{{y}_{t}}$

${{y}^{\prime }}_{t}=K+a{b}^{t}$

${s}_{1}=\underset{i=1}{\overset{n}{\sum }}{y}_{i},{s}_{2}=\underset{i=n+1}{\overset{2n}{\sum }}{y}_{i},{s}_{3}=\underset{i=2n+1}{\overset{3n}{\sum }}{y}_{i}$

$\left\{\begin{array}{l}b={\left(\frac{{s}_{3}-{s}_{2}}{{s}_{2}-{s}_{1}}\right)}^{1/n}\\ a=\left({s}_{2}-{s}_{1}\right)\frac{b-1}{b{\left({b}^{n}-1\right)}^{2}}\\ K=\frac{1}{m}\left[{s}_{1}-\frac{ab\left({b}^{n}-1\right)}{b-1}\right]\end{array}$ (5)

3. 实证分析

3.1. 指标选取

3.2. 数据分析

${{x}^{\prime }}_{i}=\frac{{x}_{i}}{\stackrel{¯}{x}}$

Table 1. Data of the main variables

3.3. 指标合理化分析

$R=\left(\begin{array}{lllll}0.5906\hfill & 0.8618\hfill & 0.6643\hfill & 0.5271\hfill & 0.6246\hfill \end{array}\right)$

1) 第4个值最小，但是也是大于0.5的，表明各种城乡差距对粮食的种植面积影响相对较小，即随着城乡收入差距的变化，并没有对农民的生活有太大影响。

2) 第2个值最大，表明农村的家庭负担对粮食的种植面积的影响很大，即当农民家庭负担重时，有的就去城市打工了，导致农田荒废，耕地面积减少。

3) 第5个值相对较大，说明城镇化率对种植面积的影响较大，即随着城镇化率的加大，房地产也随之发展，导致建房子用地增多，粮食种植面积减少。

4) 以上分析表明这五个因子对粮食的种植面积有较大影响。

3.4. 回归模型的建立

$y={\beta }_{0}+{\beta }_{1}{x}_{1}+{\beta }_{2}{x}_{2}+{\beta }_{3}{x}_{3}+{\beta }_{4}{x}_{4}+{\beta }_{5}{x}_{5}+\epsilon$

$y=1.10862-0.11304{x}_{1}+0.03855{x}_{2}-0.28897{x}_{3}+0.1005{x}_{4}+0.086655{x}_{5}$

3.5. 原模型的检验和分析

${\stackrel{^}{\beta }}_{0}=1.10862,{\stackrel{^}{\beta }}_{1}=0.1862,{\stackrel{^}{\beta }}_{2}=0.03855,{\stackrel{^}{\beta }}_{3}=0.28897,{\beta }_{4}=0.105,{\beta }_{5}=0.08358$ 相应的检验p值分别为这表明对y与 ${x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5}$ 的线性的假设不合理。

3.6. 模型的优化

$y={\beta }_{0}+{\beta }_{1}\mathrm{ln}{x}_{1}+{\beta }_{3}\mathrm{ln}{x}_{2}+{\beta }_{4}{x}_{3}^{3}+{\beta }_{5}{x}_{4}^{2}+{\beta }_{6}\mathrm{ln}{x}_{5}+\epsilon$

$y=0.78282-0.03691\mathrm{ln}{x}_{1}+0.035391\mathrm{ln}{x}_{2}+0.08796{x}_{3}^{3}-0.04228{x}_{4}^{2}+0.32479\mathrm{ln}{x}_{5}+\epsilon$

Table 2. Variance analysis

Table 3. Parameter estimation

Figure 1. The relationship between the planting area and the import and export trade of grain

Figure 2. The relationship between the planting area and the family burden

Figure 3. The relationship between the planting area and the number of labor population

Figure 4. The relationship between the area of planting and the gap between urban and rural areas

Figure 5. Relationship between the urbanization rate of planting area

3.7. 优化模型的检验分析

3.8. 模型的建立与求解

${\stackrel{^}{y}}_{t1}=1.937×{10}^{32}+3.8533×{73.7723}^{t}$

Figure 6. QQ diagram

Figure 7. The relationship between the planting area and the minimum purchase price

Figure 8. The relationship between the planting area and the minimum purchase price (× 104)

$a=5.6345×{10}^{-6},b=0.9775,K=0.0056$

${\stackrel{^}{y}}_{t1}=\frac{1}{0.0056+5.6345×{10}^{-6}×{0.9775}^{t}}$

${\stackrel{^}{y}}_{t2}=\frac{1}{0.00177+6.0854×{10}^{-6}×{0.9769}^{t}}$ (6)

$t=4824.08793-44.34314y+0.17407{y}^{2}+\epsilon$ (7)

3.9. 合理性评价

Table 4. A comparison between the actual minimum price and the estimated purchase price

4. 总结

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