Time-Related Characteristics of Tenancy Contracts and Investment in Soil Conservation Practices

Abstract

We present a dynamic model that shows how different types of land tenancy contracts and their time-related characteristics influence farmers’ decisions to invest in soil improvement and productive inputs. Using recent household and plot-level data from the Brong-Ahafo Region in Ghana, we analyze the impact of land tenancy arrangements, contract duration, as well as the number of times the contract has been renewed in the past on the intensity of investment in soil conservation measures such as ditches and farmyard manure and productive inputs like chemical fertilizer. The empirical findings generally confirm the predictions of the theoretical model and reveal that the intensity of investments on different plots cultivated by a given farmer varies significantly with the type of tenancy arrangement on the plot as well as the time-related characteristics of the contract.

Appendices

Appendix 1

The shadow value of the soil quality at time $t$ is given by

$$\begin{aligned} \lambda ^C((i-1)k+s)&= e^{\int \limits _{(i-1)k}^{(i-1)k+s} {\varphi +\alpha _\gamma Y_S dv} }\int \limits _{(i-1)k+s}^{ik} e^{-\int \limits _{(i-1)k}^{(i-1)k+s} {\varphi +\alpha _\gamma Y_S dv}}\nonumber \\&pY_S dw+\lambda ^C(ik), \quad 0 \le s\le k,\quad 1\le i\le T/k \nonumber \\ \lambda ^O(t)&= e^{\int \limits _0^t {\varphi +\alpha _\gamma Y_S dv} }\int \limits _t^T {e^{-\int \limits _0^t {\varphi +\alpha _\gamma Y_S dv} }pY_S } dw+\lambda ^O(T),\quad 0\le t\le T,\qquad \end{aligned}$$

(19)

For $\lambda ^C$ note that $t=(i-1)k+s$. An increase in soil quality at time $t$ benefits the fixed-rent farmer’s or sharecropper’s farm profits at the most for $k$-years, whereas the owner-cultivator takes advantage of the same increase until $T$. This can be seen in Eq. (14) from the upper limit of the integrals which do not form part of an exponent. Hence, one can conclude that the shadow price of the fixed-rent tenant or sharecropper, $\lambda ^C$, is less than the shadow price of the owner cultivator, $\lambda ^O$ for any identical soil quality $S$. Yet, this result cannot be extended over the entire trajectories of $\lambda ^C$ because the trajectories of the soil qualities will only coincide by chance. If the upper limits of the integral ($ik=T$) were identical one might expect that $\lambda ^O(t)$ is lower than$\lambda ^C(t)$. However, since $\lambda ^O(t)$ and $\lambda ^C(t)$ are determined by the stream of the discounted farm profits with respect to a marginal improvement of soil quality from time $t$ until the end of the planning horizon, their values depend crucially on the length of the planning horizon. Since$\lambda ^O(t)$ takes into account the next $T-t$ years and $\lambda ^C((i-1)k+s)$ considers only the next $k-s$ years, it is expected that the longer time horizon of the owner-cultivator guarantees that $\lambda ^C$, is lower than the shadow price of the owner, $\lambda ^O$ for any soil quality $S$ and any remaining planning horizon.

Appendix 2

Due to the concavity of the production function the term $Y_S$ will decrease with an increase in soil fertility and increase with a decrease in soil fertility. For fixed-rent tenant and sharecroppers we obtain that

$$\begin{aligned} \frac{\lambda ^C}{dS}&= e^{\int \limits _{(i-1)k}^{(i-1)k+s} {\varphi +\alpha _\gamma Y_S dv} }\left( {\int \limits _{(i-1)k}^{(i-1)k+s} {\alpha _\gamma Y_{SS} dv} }\right) \int \limits _{(i-1)k+s}^{ik} {e^{-\int \limits _{(i-1)k}^{(i-1)k+s} {\varphi +\alpha _\gamma Y_S dv} }pY_S } dw \\&\quad +\,e^{\int \limits _{(i-1)k}^{(i-1)k+s} {\varphi +\alpha _\gamma Y_S dv} }\left[ \int \limits _{(i-1)k+s}^{ik} {e^{-\int \limits _{(i-1)k}^{(i-1)k+s} {\varphi +\alpha _\gamma Y_S dv} }\left( {-\int \limits _{(i-1)k}^{(i-1)k+s} {\alpha _\gamma Y_{SS} dv} }\right) } \right. \\&\left. \qquad \qquad \qquad \qquad \qquad \quad + e^{-\int \limits _{(i-1)k}^{(i-1)k+s} {\varphi +\alpha _\gamma Y_S dv} }pY_{SS} dw\right] \\&= e^{\int \limits _{(i-1)k}^{(i-1)k+s} {\varphi +\alpha _\gamma Y_S dv} }\left[ {\int \limits _{(i-1)k+s}^{ik} {e^{-\int \limits _{(i-1)k}^{(i-1)k+s} {\varphi +\alpha _\gamma Y_S dv} }pY_{SS} dw} } \right] <0. \end{aligned}$$

Following the same steps we also obtain that ${d\lambda ^O}/dS<0$.

Appendix 3

In the opposite case erosion processes are dominant and cannot be compensated by the application of manure, $(\alpha _M -\alpha _Y Y_M )<0$. In this case, as shown in Fig. 4 and summarized in Observation 2 it is efficient for owners to apply less manure than fixed-rent tenants, i.e., $\tilde{M}^O<M^T$. Depending on the severity of the erosion processes, owners may even apply less manure than sharecroppers, i.e., $\tilde{\tilde{M}}^O< \quad M^S$. Owners choose $\tilde{\tilde{M}}^O$, if the marginal value of the decrease in soil fertility, $\left| {\lambda ^O(\alpha _M -\alpha _y Y_M )} \right|$ is greater than the value of the marginal product that corresponds to the sharcropper, $\gamma pY_M$. However, owners apply more manure than sharecroppers if $\left| {\lambda ^O(\alpha _M -\alpha _y Y_M )} \right| <\gamma pY_M$.

Fig. 4

Determination of the optimal amount of manure applied by sharecroppers, fixed-rent tenants and owner-cultivators for the case where $(\alpha _M -\alpha _Y Y_M)<0$

Appendix 4

To examine how the optimal investment behavior evolves over time we choose for the sake of concreteness a specific point in time, let say $k/2$, i.e., we analyze the optimal investment behavior of the different farmers for the cases where $S(\left. t \right| _{t>k/2} )>S_0$, and $S(\left. t \right| _{t>k/2} )<S_0$. If $S(\left. t \right| _{t>k/2} )>S_0$, it is optimal for farmers to build up soil fertility over time, while it would be optimal for them to reduce soil fertility, if $S(\left. t \right| _{t>k/2} )<S_0$. We consider the case where owner-cultivators build up soil fertility ($S(\left. t \right| _{t>k/2} )>S_0 )$, while tenants do the opposite. In this situation, the curves $pY_M$ and $(1-\gamma )pY_M$ rotate clockwise as shown by the discontinuous line for the case of $(1-\gamma )pY_M$ in Fig. 1. Similarly, the curve $pY_M +\lambda ^O(\alpha _M -\alpha _Y Y_M )$ rotates counter clockwise. Figure 1 shows a decline in the optimal level of manure for owner-cultivators ($\tilde{M}^O)$, and an increase in the optimal level for sharecroppers over time ($\tilde{M}^S)$. The optimal behavior of fixed-rent tenants is similar to that of sharecroppers, but is not shown in Fig. 1 in the interest of brevity. The opposite results are obtained, if owner-cultivators reduce soil fertility ($S(\left. t \right| _{t>k/2} )<S_0 )$, and tenants build up soil capital ($S(\left. t \right| _{t>k/2} )>S_0 )$. Likewise, if we consider the case where $F,M$ and $S$ are complements, i.e., $Y_{FS} ,Y_{MS} >0$, the curves would rotate clockwise, resulting in opposite results.

Appendix 5

See Table 4.

Table 4 First-stage probits estimates of determinants of tenancy arrangement