Environment and Economy in the Modern Agricultural Development

Abstract

This chapter establishes a three-sector general equilibrium model to investigate the environmental and economic effects of policies intended to promote modern agriculture. In the model, two situations are considered: in the first situation, the perfect mobility of capital between the capital-consuming sectors is assumed, and in the second situation, there is perfect mobility of land between the land-using sectors, keeping perfect mobility of capital assumption unchanged. The main conclusion is that the environmental effect of interest subsidization for modern agricultural sector is superior to other subsiding policies of factor prices.

Appendices

Appendices

1.1 Appendix A

Substituting dK ₁ ∕ ds ₁=0, dw ₂ ∕ ds ₁=w ₂>0 into Eqs. (13.1), (13.2), (13.3), (13.4), (13.5), (13.6), (13.7), (13.8), (13.9), (13.10), (13.11), (13.12), and (13.13), we get

$\frac{dL_1}{ds_1}=\frac{dE}{ds_1}=\frac{dK_2}{ds_1}=\frac{dL_2}{ds_1}=\frac{dr}{ds_1}=\frac{dX_2}{ds_1}=\frac{dX_1}{ds_1}=\frac{dw_3}{ds_1}=0$, $\frac{d\mu}{{d s}_1}>0$, $\frac{dL_3}{ds_1}<0$, $\frac{dX_3}{ds_1}<0$.

1.2 Appendix B

Substituting dK ₁ ∕ ds ₂=1 ∕ Ψ<0 into Eqs. (13.1), (13.2), (13.3), (13.4), (13.5), (13.6), (13.7), (13.8), (13.9), (13.10), (13.11), (13.12), and (13.13), we get

$\frac{dL_1}{ds_2}<0$, $\frac{dE}{ds_2}>0$, $\frac{dK_2}{ds_2}>0$, $\frac{dL_2}{ds_2}>0$, $\frac{dr}{ds_2}=0$, $\frac{dX_1}{ds_2}<0$, $\frac{dX_2}{ds_2}>0$, $\frac{dw_2}{ds_2}>0$, $\frac{dw_3}{ds_2}>0$. When${\varepsilon}_2{w}_2{L}_2>{\varepsilon}_3{w}_3\left(\overline{L}-{L}_3\right)$, dμ ∕ ds ₂>0; when ${\varepsilon}_2{w}_2{L}_2>{\varepsilon}_3{w}_3\left(\overline{L}-{L}_3\right)$ and ${k}_1/{k}_2>{\overline{w}}_1/\left(\alpha {w}_2\right)$, dL ₃ ∕ ds ₂<0; when ${\varepsilon}_2{w}_2{L}_2>{\varepsilon}_3{w}_3\overline{L}$, dX ₃/ds ₁ < 0.

1.3 Appendix C

Differentiating Eqs. (13.4), (13.5), (13.6′*), (13.7′), (13.8), (13.9′), (13.10′), (13.12), (13.13′), (13.18), (13.19), and (13.20) and writing in a matrix notation, we can obtain the following equation:

$$ \left[\begin{array}{ccccc}\hfill A\hfill & \hfill M\hfill & \hfill {\left({w}_2\right)}^{-1}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill B\hfill & \hfill N\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill C\hfill & \hfill P\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill S\hfill \\ {}\hfill D\hfill & \hfill Q\hfill & \hfill 0\hfill & \hfill {\left({w}_3\right)}^{-1}\hfill & \hfill T\hfill \\ {}\hfill G\hfill & \hfill R\hfill & \hfill {L}_2\hfill & \hfill -\left(\overline{L}-{L}_3\right)\hfill & \hfill {w}_3\hfill \end{array}\right]\left[\begin{array}{c}\hfill {dK}_1\hfill \\ {}\hfill {dT}_2\hfill \\ {}\hfill {dw}_2\hfill \\ {}\hfill {dw}_3\hfill \\ {}\hfill {dL}_3\hfill \end{array}\right]=\left[\begin{array}{c}\hfill 1\hfill \\ {}\hfill 0\hfill \\ {}\hfill 0\hfill \\ {}\hfill 0\hfill \\ {}\hfill 0\hfill \end{array}\right]{ds}_1, $$

(13.21)

where

$$ \begin{array}{l}A=\frac{\varepsilon_2\lambda {X}_1}{EK_1}+\frac{\beta {L}_2{F}_{LL}^2+{K}_2{F}_{LK}^2}{F_L^2{K}_2},\kern0.5em B=\frac{\varepsilon_2\lambda {X}_1}{EK_1}+\frac{\beta {L}_2{F}_{KL}^2+{K}_2{F}_{KK}^2}{F_K^2{K}_2},C=\frac{\left({\varepsilon}_2-{\varepsilon}_3\right)\lambda {X}_1}{EK_1}\hfill \\ {}+\frac{\beta {L}_2{F}_{TL}^2+{K}_2{F}_{TK}^2}{F_T^2{K}_2},\kern0.5em D=\frac{\varepsilon_3\lambda {X}_1}{EK_1}>0,\kern0.5em G=\frac{{\overline{w}}_1{L}_1}{K_1}\hfill \\ {}-\frac{w_2\beta {L}_2}{K_2},M=-\frac{\gamma {L}_2{F}_{LL}^2+{T}_2{F}_{LT}^2}{F_L^2{T}_2},\kern0.5em N=-\frac{\gamma {L}_2{F}_{KL}^2+{T}_2{F}_{KT}^2}{F_K^2{T}_2}<0,\hfill \\ {}\kern0.1em P=-\left(\frac{\gamma {L}_2{F}_{TL}^2+{T}_2{F}_{TT}^2}{F_T^2{T}_2}+\frac{F_{TT}^3}{F_T^3}\right)>0,Q=\frac{F_{LT}^3}{F_L^3}>0,\hfill \\ {}\kern0.1em R=\frac{w_2\beta {L}_2}{T_2}>0,\kern0.5em S=\frac{F_{TL}^3}{F_T^3}>0,\kern0.5em T=-\frac{F_{LL}^3}{F_L^3}>0.\hfill \end{array} $$

The determinant of square matrix in Eq. (13.21) is

$\Delta =-\left\{S\left[{w}_2{L}_2\left(\mathit{\mathrm{AN}}- BM\right)+\left( BR- GN\right)+\left(\overline{L}-{L}_3\right){w}_3\left( BQ- DN\right)\right]\right\}/{w}_2{w}_3+\left[1+\left(\overline{L}-{L}_3\right)T\right]\left( BP- CN\right)/{w}_2$. Since the sign of Δ cannot be directly determined, we will use dynamic adjustment to decide its sign.

Let

$$ {\dot{L}}_1={d}_1\left({P}_1{F}_L^1\left({L}_1,{K}_1\right)-{\overline{w}}_1\right), $$

(13.C1)

$$ {\dot{L}}_2={d}_2\left({P}_2{E}^{\varepsilon_2}{F}_L^2\left({L}_2,{K}_2,{T}_2\right)-{w}_2\right), $$

(13.C2)

$$ {\dot{L}}_3={d}_3\left({E}^{\varepsilon_3}{F}_L^3\left({L}_3,{T}_3\right)-{w}_3\right), $$

(13.C3)

$$ {\dot{K}}_1={d}_4\left({P}_1{F}_K^1\left({L}_1,{K}_1\right)- r\right), $$

(13.C4)

$$ {\dot{K}}_2={d}_5\left({P}_2{E}^{\varepsilon_2}{F}_K^2\left({L}_2,{K}_2,{T}_2\right)- r\right), $$

(13.C5)

$$ {\dot{T}}_2={d}_6\left({P}_2{E}^{\varepsilon_2}{F}_T^2\left({L}_2,{K}_2,{T}_2\right)-\tau \right), $$

(13.C6)

$$ {\dot{T}}_3={d}_7\left({E}^{\varepsilon_3}{F}_T^3\left({L}_3,{T}_3\right)-\tau \right), $$

(13.C7)

$$ {\dot{w}}_2={d}_8\left({L}_2-{A}_2{K_2}^{\beta}{T}_2^{\gamma}\right), $$

(13.C8)

$$ {\dot{w}}_3={d}_9\left({\overline{w}}_1{L}_1+{w}_2{L}_2-{w}_3\left(\overline{L}-{L}_3\right)\right), $$

(13.C9)

$$ \dot{r}={d}_{10}\left({K}_1+{K}_2-\overline{K}\right), $$

(13.C10)

$$ \dot{\tau}={d}_{11}\left({T}_2+{T}_3-\overline{T}\right), $$

(13.C11)

$$ \dot{E}={d}_{12}\left( E-\overline{E}+\lambda {F}^1\right), $$

(13.C12)

where “^.” represents differentiation with respect to time and d _j (j=1,2…,12) is the positive coefficient measuring the speed of adjustment and d _j >0.

The determinant of the Jacobian matrix of Eqs. (13.C1), (13.C2), (13.C3), (13.C4), (13.C5), (13.C6), (13.C7), (13.C8), (13.C9), (13.C10), (13.C11), and (13.C12) is

$$ \begin{array}{ll}\hfill & \left|J\right|\\ {}& =\left|\begin{array}{cccccccccccc}\hfill {d}_1{P}_1{F}_{LL}^1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {d}_1{P}_1{F}_{LK}^1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {d}_2{P}_2{E}^{\varepsilon_2}{F}_{LL}^2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {d}_2{P}_2{E}^{\varepsilon_2}{F}_{LK}^2\hfill & \hfill {d}_2{P}_2{E}^{\varepsilon_2}{F}_{LT}^2\hfill & \hfill 0\hfill & \hfill -{d}_2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {d}_2{P}_2{\varepsilon}_2{E}^{\varepsilon_2-1}{F}_L^2\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill {d}_3{E}^{\varepsilon_3}{F}_{LL}^3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {d}_3{E}^{\varepsilon_3}{F}_{LT}^3\hfill & \hfill 0\hfill & \hfill -{d}_3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {d}_3{\varepsilon}_3{E}^{\varepsilon_3-1}{F}_L^3\hfill \\ {}\hfill {d}_4{P}_1{F}_{KL}^1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {d}_4{P}_1{F}_{KK}^1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{d}_4\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {d}_5{P}_2{E}^{\varepsilon_2}{F}_{KL}^2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {d}_5{P}_2{E}^{\varepsilon_2}{F}_{KK}^2\hfill & \hfill {d}_5{P}_2{E}^{\varepsilon_2}{F}_{KT}^2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{d}_5\hfill & \hfill 0\hfill & \hfill {d}_5{P}_2{\varepsilon}_2{E}^{\varepsilon_2-1}{F}_K^2\hfill \\ {}\hfill 0\hfill & \hfill {d}_6{P}_2{E}^{\varepsilon_2}{F}_{TL}^2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {d}_6{P}_2{E}^{\varepsilon_2}{F}_{TK}^2\hfill & \hfill {d}_6{P}_2{E}^{\varepsilon_2}{F}_{TT}^2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{d}_6\hfill & \hfill {d}_6{P}_2{\varepsilon}_2{E}^{\varepsilon_2-1}{F}_T^2\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill {d}_7{E}^{\varepsilon_3}{F}_{TL}^3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {d}_7{E}^{\varepsilon_3}{F}_{TT}^3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{d}_7\hfill & \hfill {d}_7{\varepsilon}_3{E}^{\varepsilon_3-1}{F}_T^3\hfill \\ {}\hfill 0\hfill & \hfill {d}_8\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -{d}_8\beta \frac{L_2}{K_2}\hfill & \hfill -{d}_8\gamma \frac{L_2}{T_2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {d}_9{\overline{w}}_1\hfill & \hfill {d}_9{w}_2\hfill & \hfill {d}_9{w}_3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {d}_9{L}_2\hfill & \hfill -{d}_9\left(\overline{L}-{L}_3\right)\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {d}_{10}\hfill & \hfill {d}_{10}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {d}_{11}\hfill & \hfill {d}_{11}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {d}_{12}\lambda {F}_L^1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {d}_{12}\lambda {F}_K^1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {d}_{12}\hfill \end{array}\right|.\hfill \end{array} $$

It can also be written as follows: $\left| J\right|=-{P}_1{F}_{LL}^1 r\tau {w}_2{w}_3{d}_1{d}_2\dots {d}_{12}\Delta$. Under the condition of a stable system, there must be |J| > 0, and thus Δ >0.

1.4 Appendix D

Using Cramer’s rule to solve Eq. (13.21), we get dK ₁/ds ₁ =  − L ₂ SN/Δw ₃ > 0, dT ₂/ds ₁ = L ₂ BS/Δw ₃. Note that ${\varGamma}_{K K}={F}_{K K}^2{K}_2/{F}_K^2$, which denotes the capital elasticity of its marginal product (or interest rate) in the modern agricultural sector; ${\varGamma}_{K L}={F}_{K L}^2{L}_2/{F}_K^2$, which denotes the labor elasticity of the marginal product of capital (or interest rate) in the modern agricultural sector. We assume that |βΓ _KL +Γ _KK |< e, where βΓ _KL +Γ _KK is the resultant capital elasticity of interest rate in the modern agricultural sector,e = ε ₂ λK ₂ X ₁/EK ₁. Because we separate land factor from capital, it makes the interest rate less sensitive to capital factor’ s change. Hence, B > 0 and dT ₂ ∕ ds ₁>0.

Substituting the above results into Eqs. (13.1′), (13.2′), (13.4), (13.8), (13.10′), (13.12), (13.18), and (13.20), we can obtain the following:

$$ \begin{array}{l}\frac{dK_2}{ds_1}<0,\kern0.5em \frac{dT_3}{ds_1}<0,\kern0.5em \frac{dr}{ds_1}=0,\kern0.5em \frac{dE}{ds_1}<0,\kern0.5em \frac{dX_1}{ds_1}>0,\\ {}\frac{dL_2}{ds_1}=-\frac{L_2{SL}_2}{\Delta {w}_3}\left[\frac{\gamma {K}_2{F}_{KK}^2-\beta {T}_2{F}_{KT}^2}{F_K^2{K}_2{T}_2}+\frac{{\gamma \varepsilon}_2\lambda {X}_1}{T_2{EK}_1}\right],\\ {}\frac{dX_2}{ds_1}=\frac{L_2S}{\Delta {w}_3}\left[\frac{\varepsilon_2\lambda {X}_1{X}_2N}{EK_1}+ rN-{w}_2{L}_2\left(\frac{\gamma {K}_2{F}_{KK}^2-\beta {T}_2{F}_{KT}^2}{F_K^2{K}_2{T}_2}+\frac{{\gamma \varepsilon}_2\lambda {X}_1}{T_2{EK}_1}\right)\right].\end{array} $$

1.5 Appendix E

Using Cramer’s rule to solve Eq. (13.22), we get

$\frac{dK_1}{ds_3}=\frac{- N\left[\left(\overline{L}-{L}_3\right) T+1\right]}{\Delta {w}_2}>0,$ $\frac{dT_2}{ds_3}=\frac{B\left[\left(\overline{L}-{L}_3\right) T+1\right]}{\Delta {w}_2}>0.$

Substituting the results of Eq. (13.22) into Eqs. (13.1′), (13.2′), (13.4), (13.8), (13.10′), (13.12), (13.18), and (13.20), we obtain the following:

$$ \begin{array}{l}\frac{{d K}_2}{{d s}_3}<0,\frac{{d T}_3}{{d s}_3}<0,\kern0.5em \frac{d r}{{d s}_3}=0,\kern0.5em \frac{d E}{{d s}_3}<0,\kern0.5em \frac{{d X}_1}{{d s}_3}>0,\kern0.5em \frac{d\tau}{{d s}_3}<0,\\ {}\frac{{d L}_2}{{d s}_3}=\left(\frac{\beta N}{K_2}+\frac{\gamma B}{T_2}\right)\frac{L_2\left[\left(\overline{L}-{L}_3\right)T+1\right]}{\Delta {w}_2},\\ {}\frac{{d X}_2}{{d s}_3}=\left[\frac{\varepsilon_2\lambda {X}_1{X}_2N}{EK_1}+ rN+{w}_2{L}_2\left(\frac{\beta N}{K_2}+\frac{\gamma B}{T_2}\right)\right]\frac{\left[\left(\overline{L}-{L}_3\right)T+1\right]}{\Delta {w}_2}\end{array} $$