Contract design for technology sharing between two farmers

Abstract

Technology sharing among farmers has become common but controversial in recent years. To address it, we consider two farmers engaged in Cournot competition to investigate motivations of technology sharing and provide suggestions on contract design. One farmer (a licensor) has developed some technology and decides whether to share technology with the other farmer (a licensee). The licensee chooses whether to buy technology under a fixed-rate contract or a royalty-fee contract. We propose a technology sharing ratio between two farmers to characterize the degree of technology sharing. We find a win–win outcome for both farmers when the technology sharing ratio is higher than a threshold under the fixed-fee contract. While under the royalty-fee contract, the licensor only shares technology with an additional constraint that they have similar production costs. When the licensor can design contracts, he prefers the royalty-fee contract to the fixed-fee contract. We further interpret why the licensor may not benefit more under the two-part tariff contract than the fixed-fee or the royalty-fee contract. Moreover, we find that in supply chain settings, a win–win outcome for both farmers exists if and only if the technology sharing ratio is smaller than a threshold under the fixed-fee contract while technology sharing will not be realized under the royalty-fee contract. Finally, we show that the strategy of whether to share technology is robust to yield uncertainty, and both the licensor and licensee may benefit more from technology sharing because of yield uncertainty.

Appendices

Appendix

See Tables 2, 3 and 4.

Proof of Table 1

When technology sharing is not achieved, since $\partial^{2} \pi_{i} /\partial q_{i}^{2} = - 2 < 0$, $\pi_{i}$ is a concave function of $q_{i}$. Hence, solving the first-order conditions $\partial \pi_{1} /\partial q_{1} = 0$ and $\partial \pi_{2} /\partial q_{2} = 0$, we have $q_{1}^{B*} = \left( {a - 2c_{1} + c_{2} } \right)/3$ and $q_{2}^{B*} = \left( {a + c_{1} - 2c_{2} } \right)/3$. Then we get $p^{B*}$, $\pi_{1}^{B*}$, and $\pi_{2}^{B*}$ from Eqs. (2), and (3). Plugging $q_{i}^{B*}$ and $p^{B*}$ into Eq. (1), we can get $CS_{i}^{B*}$. As shown in the second column of Table 1.

When technology sharing is achieved under the FF contract or the RF contract between Farmers 1 and 2, proofs are similar to that without technology sharing.

Proof of Proposition 1

Under the RF contract, to guarantee $\partial \pi_{1}^{RF*} /\partial r = \left[ {5a - (1 + 4k)c_{1} - (4 - 4k)c_{2} - 10r} \right]/9 < 0$, we have $r > \left( {5a - c_{1} - 4c_{2} - 4kc_{1} + 4kc_{2} } \right)/10$. To guarantee $\partial \pi_{2}^{RF*} /\partial r = [(1 - 2k)c_{1} - 2(1 - k)c_{2} + a - 4r]/3 > 0$, we have $r > \left( {a + c_{1} - 2c_{2} - 2kc_{1} + 2kc_{2} } \right)/2$. Since $\left( {a + c_{1} - 2c_{2} - 2kc_{1} + 2kc_{2} } \right)/2 > \left( {5a - c_{1} - 4c_{2} - 4kc_{1} + 4kc_{2} } \right)/10$, we have Proposition 1.

Proof of Proposition 2

To guarantee $\pi_{1}^{FF*} - \pi_{1}^{RF*} = \left[ {r\left( {4c_{2} + c_{1} - 5a + 4kc_{1} - 4kc_{2} + 5r} \right) + 9kF} \right]/9 > 0$, we have $F > \left( {5a - c_{1} - 4c_{2} - 4kc_{1} + 4kc_{2} - 5r} \right)r/\left( {9k} \right)$. To guarantee $\pi_{2}^{{FF{*}}} - \pi_{2}^{{RF{*}}} = \left[ {\left( {a - 2c_{1} + kc_{1} + c_{2} - kc_{2} } \right)^{2} - \left( {a + c_{1} - 2kc_{1} - 2c_{2} + 2kc_{2} - 2r} \right)^{2} + 9kF} \right]/9 > 0$, we have $F < 4\left( {a + c_{1} - 2c_{2} - 2kc_{1} + 2kc_{2} - r} \right)r/\left( {9k} \right)$. To guarantee $\left( {5a - c_{1} - 4c_{2} - 4kc_{1} + 4kc_{2} - 5r} \right)r/\left( {9k} \right) < 4\left( {a + c_{1} - 2c_{2} - 2kc_{1} + 2kc_{2} - r} \right)r/\left( {9k} \right)$, we have $\max \left\{ {r/\left( {c_{2} - c_{1} } \right),\left( {c_{2} - c_{1} + r} \right)/\left( {2c_{2} - 2c_{1} } \right)} \right\} < k < 1$. Hence, we have Proposition 2.

Proof of Proposition 3

Under the FF contract, Farmer 1 shares his technology when $\pi_{1}^{FF*} > \pi_{1}^{B*}$, Hence, solving the inequality $\pi_{1}^{FF*} - \pi_{1}^{B*} = \left[ {\left( {c_{2} - c_{1} } \right)\left( {4c_{1} - kc_{1} - 2c_{2} + kc_{2} - 2a} \right) + 9F} \right]/9 > 0$, we have $F \ge F_{1}^{FF} = (c_{2} - c_{1} )\left[ {2a - c_{1} (4 - k) + c_{2} (2 - k)} \right]/9 > 0$. By solving the inequality $\pi_{2}^{FR*} - \pi_{2}^{B*} = \left[ {4\left( {c_{2} - c_{1} } \right)\left( {c_{1} - kc_{1} - 2c_{2} + kc_{2} + a} \right) - 9F} \right]/9 > 0$, we have $F \le F_{2}^{FF} = 4(c_{2} - c_{1} )\left[ {a + c_{1} (1 - k) - c_{2} (2 - k)} \right]/9$. To guarantee $F_{1}^{FF} < F_{2}^{FF}$, we have {$c_{2} \ge \left( {a + c_{1} } \right)/2$ and $k > \left( {2c_{2} - c_{1} - a} \right)/\left( {c_{2} - c_{1} } \right)$} or $c_{2} < \left( {a + c_{1} } \right)/2$. Then we have Proposition 3.

Proof of Proposition 4

Similar to the proof of Proposition 3, by solving the inequality group. $\left\{ \begin{array}{l} \pi_{1}^{RF*} - \pi_{1}^{B*} = ({c_{2} - c_{1} } )(k^{2} c_{2} - k^{2} c_{1} - 2ak + 4kc_{1} -2kc_{2}\\ \quad - 5ar + rc_{1} + 4rc_{2} + 4krc_{1} - 4krc_{2} +5r^{2} )/9 > 0\\ \pi_{2}^{RF*} - \pi_{2}^{B*} = 4( {kc_{2} - kc_{1}- r} )({c_{1} - kc_{1} - 2c_{2} + kc_{2} + a - r})/9 > 0\\ \end{array} \right.$, we have Proposition 4.

Proof of Proposition 5

Under the FF contract, since $\partial^{2} \pi_{1}^{FF*} /\partial k^{2} = 2\left( {c_{1} - c_{2} } \right)^{2} /9 > 0$, $\partial^{2} \pi_{2}^{FF*} /\partial k^{2} = 8\left( {c_{1} - c_{2} } \right)^{2} /9 > 0$, i.e., $\pi_{i}^{FR*}$ is a convex function of $k$. The first-order conditions of $k$ are $\partial \pi_{1}^{FF*} /\partial k = 2\left( {c_{2} - c_{1} } \right)\left[ {\left( {2 - k} \right)c_{1} + kc_{2} - a} \right]/9 + F = 0$ and $\partial \pi_{2}^{FF*} /\partial k = 4(c_{2} - c_{1} )\left[ {(1 - 2k)c_{1} + 2(k - 1)c_{2} + a} \right]/9 - F = 0$. Then we can get $k_{1}^{FF*} = \left[ {4c_{1}^{2} + 2a(c_{2} - c_{1} ) - 6c_{1} c_{2} + 2c_{2}^{2} - 9F} \right]/\left[ {2\left( {c_{2} - c_{1} } \right)^{2} } \right]$ and $k_{2}^{FF*} = \left[ {4c_{1}^{2} + 4a(c_{1} - c_{2} ) - 12c_{1} c_{2} + 8c_{2}^{2} { + }9F} \right]/\left[ {8\left( {c_{2} - c_{1} } \right)^{2} } \right]$ corresponding to the minimum profits of Farmers 1 and 2. According to $\left\{ {\begin{gathered} \pi_{1}^{FF*} - \pi_{1}^{B*} = \left[ {\left( {c_{2} - c_{1} } \right)\left( {4c_{1} - kc_{1} - 2c_{2} + kc_{2} - 2a} \right) + 9F} \right]/9 > 0 \hfill \\ \pi_{2}^{FF*} - \pi_{2}^{B*} = \left[ {4\left( {c_{2} - c_{1} } \right)\left( {c_{1} - kc_{1} - 2c_{2} + kc_{2} + a} \right) - 9F} \right]/9 > 0 \hfill \\ \end{gathered}} \right.$, we can get that.

$k > \max \{ \frac{{4c_{1}^{2} + 2a\left( {c_{2} - c_{1} } \right) - 6c_{1} c_{2} + 2c_{2}^{2} - 9F}}{{\left( {c_{2} - c_{1} } \right)^{2} }},\frac{{4c_{1}^{2} + 4a\left( {c_{2} - c_{1} } \right) - 12c_{1} c_{2} + 8c_{2}^{2} - 9F}}{{4\left( {c_{2} - c_{1} } \right)^{2} }}\}$. Since $k_{i}^{FR*} < \min \{ \frac{{4c_{1}^{2} + 2a\left( {c_{2} - c_{1} } \right) - 6c_{1} c_{2} + 2c_{2}^{2} - 9F}}{{\left( {c_{2} - c_{1} } \right)^{2} }},\frac{{4c_{1}^{2} + 4a\left( {c_{2} - c_{1} } \right) - 12c_{1} c_{2} + 8c_{2}^{2} - 9F}}{{4\left( {c_{2} - c_{1} } \right)^{2} }}\}$, we have $\partial \pi_{1}^{{FF{*}}} /\partial k > 0$ and $\partial \pi_{2}^{{FF{*}}} /\partial k > 0$.

Under the RF contract, in the same way, we can get $\partial \pi_{1}^{{RF{*}}} /\partial k < 0$ and $\partial \pi_{2}^{{RF{*}}} /\partial k > 0$.

Proof of Proposition 6

$CS^{FF*} - CS^{B*} = \left( {c_{2} - c_{1} } \right)\left[ {4a - (2 + k)c_{1} - \left( {2 - k} \right)c_{2} } \right]k/18$, since $a > c_{2} > c_{1}$, we have $c_{2} - c_{1} > 0$ and $(2 + k)c_{1} + \left( {2 - k} \right)c_{2} < 4c_{2}$. Hence, we have $4a - (2 + k)c_{1} - \left( {2 - k} \right)c_{2} > 0$ and $CS^{FF*} - CS^{B*} > 0$. In the same way, we can get $CS^{RF*} - CS^{B*} > 0$ and $CS^{FF*} - CS^{RF*} > 0$. Then, we have Proposition 6.

Proof of Proposition 7

Under the FF contract, when Farmer 1 decides $F$, the problem of Farmer 1 is,

$$ \begin{gathered} \mathop {\max }\limits_{F} \overline{\pi }_{1}^{FF} = \left( {\overline{p} - c_{1} } \right)\overline{q}_{1} + kF \hfill \\ s.t.\;\overline{\pi }_{2}^{FF} - \overline{\pi }_{2}^{B} \ge 0 \hfill \\ \end{gathered} $$

We get $F^{*} = F_{2}^{FF}$. Under the RF contract, when Farmer 1 decides $r$, in the same way, we have $r^{*} = r_{2}^{RF}$. According to the derivation above, the profit functions of Farmers 1 and 2 are $\overline{\pi }_{1}^{{FF{*}}} { = }\left[ {\left( {a - c_{1} + 2c_{2} } \right)^{2} - 2\left( {c_{2} - c_{1} } \right)\left( {5c_{2} - 4c_{1} - a} \right)k + 5\left( {c_{2} - c_{1} } \right)^{2} k^{2} } \right]/9$ and $\overline{\pi }_{2}^{{FF{*}}} { = }\left( {a + c_{1} - 2c_{2} } \right)^{2} /9$ under the FF contract and $\overline{\pi }_{1}^{{RF{*}}} { = }\left[ {\left( {a - c_{1} + 2c_{2} } \right)^{2} - 3\left( {c_{2} - c_{1} } \right)\left( {2c_{2} - c_{1} - a} \right)k} \right]/9$ and $\overline{\pi }_{2}^{{RF{*}}} { = }\left( {a + c_{1} - 2c_{2} } \right)^{2} /9$ under the RF contract. We have $\overline{\pi }_{1}^{{RF{*}}} - \overline{\pi }_{1}^{{FF{*}}} = k\left( {c_{2} - c_{1} } \right)\left[ {\left( {1 - k} \right)5c_{1} - \left( {4 - 5k} \right)c_{2} - a} \right]/9 > 0$. Thus we have Proposition 7.

Proof of Corollary 1

Under the FF contract, when Farmer 1 designs the contract, to guarantee $\overline{\pi }_{1}^{{FF{*}}} - \overline{\pi }_{1}^{{B{*}}} = \left[ {5\left( {c_{2} - c_{1} } \right)^{2} k^{2} - 2\left( {c_{2} - c_{1} } \right)\left( {5c_{2} - 4c_{1} - a} \right)k} \right]/9 > 0$, we have $\max \left\{ {2\left( {5c_{2} - 4c_{1} - a} \right)/\left( {5c_{2} - 5c_{1} } \right),0} \right\} \le k < \min \left\{ {2\left( {5c_{2} - 4c_{1} - a} \right)/\left( {5c_{2} - 5c_{1} } \right),1} \right\}$.

Proof of Corollary 2

When Farmer 1 designs the contract, $\overline{\pi }_{1}^{{FF{*}}}$ is a convex function of $k$ due to $\partial^{2} \overline{\pi }_{1}^{{FF{*}}} /\partial k^{2} = 10\left( {c_{1} - c_{2} } \right)^{2} /9 > 0$. Given $\partial \overline{\pi }_{1}^{{FF{*}}} /\partial k = 0$, we have $k_{1}^{*} = \left( {5c_{2} - 4c_{1} - a} \right)/\left( {5c_{2} - 5c_{1} } \right)$. Since $k^{*} < 0$, we have $\partial \overline{\pi }_{1}^{{FF{*}}} /\partial k > 0$.

Proof of Corollary 3

The proof of Corollary 3 is similar to that of Proposition 5.

Proof of Proposition 8

Under the TT contract, since $\partial^{2} \pi_{i} /\partial q_{i}^{2} = - 2 < 0$, $\pi_{i}$ is a concave function of $q_{i}$. Hence, by solving the first-order conditions $\partial \pi_{1} /\partial q_{1} = 0$ and $\partial \pi_{2} /\partial q_{2} = 0$, we have $q_{1}^{TT*} = \left[ {a + (k - 2)c_{1} + (1 - k)c_{2} + r} \right]/3$ and $q_{2}^{TT*} = \left[ {a + (1 - 2k)c_{1} + 2(k - 1)c_{2} - 2r} \right]/3$. Then we get the selling price $p^{TT*} = \left[ {a + (1 + k)c_{1} + (1 - k)c_{2} + r} \right]/3$ from Eq. (2). Thus, we have $\pi_{1}^{TT*} = [a + (1 - 2k)c_{1} + (2k - 2)c_{2} - 2r]^{2} /9 + kF$ and $\pi_{2}^{TT*} = [2a - (1 + k)c_{1} + (k - 1)c_{2} + r]^{2} /18 - kF$ from Eqs. (4) and (5). To guarantee $\pi_{1}^{TT*} - \pi_{1}^{FF*} = \left( {5a - c_{1} - 4kc_{1} - 4c_{2} + 4kc_{2} - 5r} \right)r/9 > 0$, we have $c_{2} < \left( {a + c_{1} } \right)/2$. We always have $\pi_{1}^{TT*} - \pi_{1}^{RF*} = kF > 0$.

Proof of Proposition 9

Solving the inequality group

$\left\{ \begin{array}{l} \pi_{1}^{TT*} - \pi_{1}^{B*} = ({c_{2} - c_{1} } )( k^{2} c_{2} - k^{2} c_{1} - 2ak + 4kc_{1} -2kc_{2}\\ \quad - 5ar + rc_{1} + 4rc_{2} + 4krc_{1} - 4krc_{2} +5r^{2} )/9 + kF > 0\\ \pi_{2}^{TT*} - \pi_{2}^{B*} = 4( {kc_{2} -kc_{1} - r})( {c_{1} - kc_{1} - 2c_{2} + kc_{2} + a - r} )/9 - kF > 0\\ \end{array} \right.$

Similar to the proof of Proposition 3, we have $c_{2} > \left( {a + c_{1} } \right)/2$, $\max \{ 0,a + c_{1} - 2c_{2} - kc_{1} + kc_{2} \} < r < k\left( {c_{2} - c_{1} } \right)$ and $F_{1}^{TT} < F < F_{2}^{TT}$.

Proof of Corollary 4

Under the TT contract, since $\partial^{2} \pi_{i}^{TT*} /\partial k^{2} > 0$, $\pi_{i}$ is a convex function of $k$. The corresponding first-order conditions of $k$ for Farmers 1 and 2 are given by $\partial \pi_{1}^{TT*} /\partial k = 2\left( {c_{2} - c_{1} } \right)\left[ {\left( {2 - k} \right)c_{1} + kc_{2} - a + 2r} \right]/9 + F = 0$ and $\partial \pi_{2}^{TT*} /\partial k = \left[ {\left( {1 - 2k} \right)c_{1} + 2\left( {k - 1} \right)c_{2} + a - 2r} \right]^{2} /9 - F = 0$. Then we can get $k_{1}^{TT*} = 2c_{1} - c_{2} - a + 2r/(c_{2} - c_{1} )$ and $k_{2}^{TT*} = 2c_{2} - c_{1} - a + 2r/\left( {2c_{2} - 2c_{1} } \right)$ corresponding to the minimum profits of Farmers 1 and 2. We find that $k_{1}^{TT*} > 1$ and $k_{2}^{TT*} < \min \{ 0,a + c_{1} - 2c_{2} - kc_{1} + kc_{2} \}$. Hence, $\partial \pi_{1}^{{TT{*}}} /\partial k < 0$, and $\partial \pi_{2}^{{TT{*}}} /\partial k > 0$.

Proof of Table 4

When technology sharing is not achieved in supply chain settings, using backward induction technique, since $\partial^{2} \pi_{ri} /\partial q_{i}^{2} { = } - 2 < 0$, $\pi_{ri}$ is a concave function of $q_{i}$. Hence, solving the first-order conditions $\partial \pi_{r1} /\partial q_{1} = 0$ and $\partial \pi_{r2} /\partial q_{2} = 0$, we have $q_{1} (w_{1} ,w_{2} ) = \left( {a - 2w_{1} + w_{2} } \right)/3$ and $q_{2} (w_{1} ,w_{2} ) = \left( {a + 2w_{1} - w_{2} } \right)/3$. Since $\partial^{2} \pi_{i} /\partial w_{i}^{2} { = } - 4/3 < 0$, $\pi_{i}$ is a concave function of $w_{i}$. Bringing $q_{i} (w_{1} ,w_{2} )$ into Eqs. (7) and (8), we solve the first-order conditions $\partial \pi_{1} /\partial w_{1} = 0$ and $\partial \pi_{2} /\partial w_{2} = 0$, and have $w_{1}^{B*} = \left( {5a{ + }8c_{1} + 2c_{2} } \right)/15$ and $w_{2}^{B*} = \left( {5a{ + 2}c_{1} + 8c_{2} } \right)/15$. Then, we have $q_{1}^{{B{*}}} = \left( {10a - 14c_{1} + 4c_{2} } \right)/45$ and $q_{2}^{{B{*}}} = \left( {10a{ + }4c_{1} - 14c_{2} } \right)/45$. Next, introducing $q_{i}^{{B{*}}}$ and $w_{i}^{{B{*}}}$ into Eqs. (2), and (6)–(8), we have $p_{i}^{{B{*}}}$, $\pi_{ri}^{{B{*}}}$ and $\pi_{i}^{{B{*}}}$. Under the FF contract and the RF contract in supply chain settings, proofs are similar to that when technology sharing is not achieved.

Table 4 The optimal decision and profit in supply chain settings

Proof of Proposition 10

Under the FF contract in supply chain settings, if $\pi_{1}^{FF*} > \pi_{1}^{B*}$, Farmer 1 shares technology. If $\pi_{2}^{FF*} > \pi_{2}^{B*}$, Farmer 2 will accept technology sharing. Hence, solving the inequalities.

$$\left\{ {\begin{array}{l} \pi_{1}^{FF*} - \pi_{1}^{B*} = \left[ {2\left( {2kc_{1} - 7c_{1} - 2kc_{2} + 2c_{2} + 5a} \right)^{2} - 2\left( {2c_{2} - 7c_{1} + 5a} \right)^{2} } \right]/675 + 9F > 0 \hfill \\ \pi_{2}^{FF*} - \pi_{2}^{B*} = k\left[ {\left( {64 - 322k} \right)c_{1}^{2} + \left( {224 - 322k} \right)c_{2}^{2} + 160a\left( {c_{2} - c_{1} } \right) + 4\left( {161k - 72} \right)c_{1} c_{2} - 675F} \right]/675 > 0 \hfill \\ \end{array}} \right.$$

Similar to the proof of Proposition 3, we have Proposition 10.

Proof of Proposition 11

Under the FF contract in supply chain settings, similar to the proof of Proposition 4, solving the inequality group.$\left\{ \begin{array}{l} \pi_{1}^{RF*} - \pi_{1}^{B*} = [3r( {65a- 66r - c_{1} - 64c_{2} } ) + 8k( {c_{2} - c_{1} } )(2c_{2} -7c_{1}\\ \quad - 24r + 5a) + 8k^{2}( {c_{2} - c_{1} })^{2} ]/675> 0\\ \pi_{2}^{RF*} - \pi_{2}^{B*} = [ 24r( {7c_{2} - 2c_{1} + 3r -5a} )+ 2k( {c_{2} - c_{1} } )( 7c_{2} - 2c_{1}\\ \quad + 6r - 5a) -112k^{2} ( {c_{2} - c_{1} })^{2} ]/675> 0\\ \end{array} \right.$, we have Proposition 11.

Proof of Footnote 2

Suppose there exists yield uncertainty. Under the FF contract, since $\partial^{2} \pi_{i} /\partial q_{i}^{2} = - 2\mu < 0$, $\pi_{i}$ is a concave function of $q_{i}$, by solving the first-order conditions $\partial \pi_{1} /\partial q_{1} = 0$ and $\partial \pi_{2} /\partial q_{2} = 0$, we have $q_{1}^{B*} = \frac{{\mu \left[ {c_{2} \mu + a\left( {\mu^{2} + 2\sigma^{2} } \right)} \right] - 2c_{1} \left( {\mu^{2} + \sigma^{2} } \right)}}{{3\mu^{4} + 8\mu^{2} \sigma^{2} + 4\sigma^{4} }}$ and $q_{2}^{B*} = \frac{{c_{1} \mu^{2} - 2c_{2} \left( {\mu^{2} + \sigma^{2} } \right) + a\mu \left( {\mu^{2} + 2\sigma^{2} } \right)}}{{3\mu^{4} + 8\mu^{2} \sigma^{2} + 4\sigma^{4} }}$. Then, plug $q_{1}^{B*}$ and $q_{2}^{B*}$ into Eqs. (9) and (10), we get $p^{B*}$, $\pi_{1}^{B*}$ and $\pi_{2}^{B*}$. Similarly, we can get $q_{1}^{FF*}$, $q_{2}^{FF*}$, $p_{{}}^{FF*}$, $\pi_{1}^{FF*}$ and $\pi_{2}^{FF*}$.

To guarantee $\pi_{1}^{FF*} > \pi_{1}^{B*}$ and $\pi_{2}^{FF*} > \pi_{2}^{B*}$, we have $F_{6}^{FF} < F < F_{5}^{FF}$.

The proof of condition for technology sharing with yield uncertainty under the RF contract is similar.

Proof of Proposition 12

Under the FF contract, we have $\Delta \pi_{1}^{FF*} = \frac{{Fk\left( {1 + 2\sigma^{2} } \right)\left( {3 + 2\sigma^{2} } \right)^{2} + \left[ {a + c_{2} - c_{1} \left( {2 - k} \right) - c_{2} k + 2\left( {a - c_{1} } \right)\sigma^{2} } \right]^{2} - \left[ {a + c_{2} + 2a\sigma^{2} - 2c_{1} \left( {1 + \sigma^{2} } \right)} \right]^{2} }}{{\left( {1 + 2\sigma^{2} } \right)\left( {3 + 2\sigma^{2} } \right)^{2} }}$

By solving the first-order condition of $\Delta \pi_{1}^{FF*}$ with respect to $\sigma$, we obtain $\frac{{\partial \Delta \pi_{1}^{FF*} }}{\partial \sigma } = \frac{{4k\sigma \left( {c_{2} - c_{1} } \right)\left\{ {4a\left( {1 + 2\sigma^{2} } \right)^{2} + c_{2} \left( {2 - k} \right)\left( {5 + 6\sigma^{2} } \right) + c_{1} \left[ {k\left( {5 + 6\sigma^{2} } \right) - 2\left( {7 + 14\sigma^{2} + 8\sigma^{4} } \right)} \right]} \right\}}}{{\left( {1 + 2\sigma^{2} } \right)^{2} \left( {3 + 2\sigma^{2} } \right)^{3} }} > 0$, then we have Proposition 12 (1).

The proof of monotonic results of Farmer 2 with yield uncertainty is similar.

The impact of the variance of yield uncertainty

Denote $\Delta \pi_{1}^{RF*} = \pi_{1}^{RF*} - \pi_{1}^{B*}$ and $\Delta \pi_{2}^{RF*} = \pi_{2}^{RF*} - \pi_{2}^{B*}$, we mainly use the following parameters: $c_{1} = 0.1$, $c_{2} = 0.8$, $a = 1$, $r = 0.1$ to show the impacts of yield uncertainty on farmers’ profits under the RF contract, as shown in Fig. 9.

Fig. 9

Impacts of yield uncertainty on farmers’ profits