Coordination for contract farming supply chain with stochastic yield and demand under CVaR criterion

Abstract

This paper analyzes the optimal production and pricing decisions in an agricultural supply chain formed by contract farming scheme consisting of an agribusiness firm and multiple risk-averse farmers. The effects of yield and demand uncertainties and the farmer’s risk aversion on the optimal decisions of the production quantity, wholesale price and retail price are analyzed. Our analyses provide managerial insights on the contract terms of the agricultural supply chain. We show that the production quantity decreases as the farmer is more risk-averse and faces higher yield uncertainty, while the retail price subsequently increases. However, the wholesale price is influenced by the interaction effect of the farmer’s risk aversion and yield uncertainty. The retail price is influenced by the interaction effect of demand uncertainty and price elasticity. In particular, we show that the loss due to the decentralized decisions increases as the farmer is more risk-averse and yield uncertainty is higher. Thus, a RPG (Revenue sharing + Production cost sharing + Guaranteed money) mechanism is developed to facilitate the coordination of the agricultural supply chain under uncertainty environment with risk-averse agents based on contract farming practices. The cost allocation ratio of the RPG mechanism borne by the agribusiness firm increases in the yield and market demand uncertainties and decreases in the farmer’s risk aversion. Specially, if the farmer is extremely risk averse, as well as the yield and demand becomes extremely higher, the RPG mechanism cannot achieve perfect coordination of the agricultural supply chain.

Appendices

Appendix 1

Proof for Theorem 1

Solve the problem $\max_{v \in R} CVaR_{\eta } (q,v)$ first. For any given $q$, it can be simplified in the following three cases.

• (1) If $v \le wqA - c(q)$, then $CVaR_{\eta } (q,v) = v$, hence ${{\partial CVaR_{\eta } (q,v)} \mathord{\left/ {\vphantom {{\partial CVaR_{\eta } (q,v)} {\partial v}}} \right. \kern-0pt} {\partial v}} = 1 > 0$.

• (2) If $wqA - c(q) < v \le wqB - c(q)$, then

  $$ {\kern 1pt} {\kern 1pt} CVaR_{\eta } (q,v) = v - \frac{1}{\eta }\int_{A}^{{\frac{v + c(q)}{wq}}} {[v - wq\mu + c(q)]} dG(\mu ) $$

  Hence

  $$ \frac{{\partial CVaR_{\eta } (q,v)}}{\partial v} = 1 - \frac{1}{\eta }G\left( {\frac{v + c(q)}{wq}} \right) $$

  Note that

  $$ \frac{{\partial CVaR_{\eta } (q,v)}}{\partial v}\left| {_{v = wAq - c(q)} } \right. = 1 > 0;\;\frac{{\partial CVaR_{\eta } (q,v)}}{\partial v}\left| {_{v = wBq - c(q)} } \right. = 1 - \frac{1}{\eta } < 0 $$

  That is, there is an optimal value $v^{*}$ to satisfy ${{\partial CVaR_{\eta } (q,v)} \mathord{\left/ {\vphantom {{\partial CVaR_{\eta } (q,v)} {\partial v}}} \right. \kern-0pt} {\partial v}} = 0$.

• (3) If $v \ge wqB - c(q)$, then ${\kern 1pt} CVaR_{\eta } (q,v) = v - \frac{1}{\eta }\int_{A}^{B} {[v - w \cdot q\mu + c(q)]dG(\mu )}$, and

  $$ \frac{{\partial CVaR_{\eta } (q,v)}}{\partial v} = 1 - \frac{1}{\eta } < 0 $$

  Hence, the optimal value $v^{*}$ is

  $$ v^{*} = wqG^{ - 1} (\eta ) - c(q) $$

  Substituting $v^{*}$ into the farmer’s objective function (4), we have

  $$ {\kern 1pt} {\kern 1pt} CVaR_{\eta } (q,v^{*} ) = v^{*} - \frac{1}{\eta }\int_{A}^{{\frac{{v^{*} + c(q)}}{wq}}} {[v^{*} - wq\mu + c(q)]} dG(\mu ) = \frac{1}{\eta }\int_{A}^{{G^{ - 1} (\eta )}} {wq\mu } dG(\mu ) - c(q) $$

  It can be easily proved that it is the strictly concave function of production quantity $q$, let ${{{\kern 1pt} {\kern 1pt} dCVaR_{\eta } (q,v^{*} )} \mathord{\left/ {\vphantom {{{\kern 1pt} {\kern 1pt} dCVaR_{\eta } (q,v^{*} )} {dq}}} \right. \kern-0pt} {dq}} = 0$, the optimal production quantity is

  $$ q_{DF}^{*} = \frac{{w\kappa_{1} (\eta ) - c_{1} }}{{2c_{2} }} $$

  where $\kappa_{1} (\eta ) = \frac{1}{\eta }\int_{A}^{{G^{ - 1} (\eta )}} \mu dG(\mu )$.□

Proof for Theorem 2

From (9) we have, the optimal value for stocking factor $z_{0}$ should satisfy the following first order partial derivative condition

$$ \frac{{\partial \varPi_{DE} (z,w){\kern 1pt} }}{\partial z}{\kern 1pt} = \frac{{y_{0}^{{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0pt} k}}} (nq\mu )^{{1 - {1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0pt} k}}} }}{{z^{{2 - {1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0pt} k}}} k}} \cdot \varPhi (z) $$

where $\varPhi (z) = z - (1 - s)\int_{\alpha }^{\text{z}} { [z + (k - 1 )x]} f(x)dx$.

Note that ${{y_{0}^{{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0pt} k}}} (nq\mu )^{{1 - {1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0pt} k}}} } \mathord{\left/ {\vphantom {{y_{0}^{{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0pt} k}}} (nq\mu )^{{1 - {1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0pt} k}}} } {(z^{{2 - {1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0pt} k}}} k)}}} \right. \kern-0pt} {(z^{{2 - {1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0pt} k}}} k)}} > 0$, then the optimal $z_{0}$ will be determined by $\varPhi (z)$. Such a $z_{0}$ always exists in the support interval $[\alpha {\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \beta ]$ of $F( \cdot )$, since ① $\varPhi (z)$ is continuous, and ② $\varPhi (\alpha ) = \alpha \ge 0$ and $\varPhi (\beta ) = s\beta - (1 - s)(k - 1)\bar{\varepsilon }$. Since

$$ \varPhi '(z) = 1 - (1 - s)[F(z) + kzf(z)] $$

$$ \varPhi ''(z) = - (1 - s)[f(z) + kf(z) + kzf'(z)] $$

Now, if $h(x) = {{xf(x)} \mathord{\left/ {\vphantom {{xf(x)} {\bar{F}(x)}}} \right. \kern-0pt} {\bar{F}(x)}}$ (where $\bar{F}(x) = 1 - F(x)$) increases in $x$, then the random variable of the market demand, $\varepsilon$, has the properties of Increasing Generalized Failure Rate (IGFR), then $\varPhi ''(z) < 0$, which implies that $\varPhi (z)$ itself is a concave function. As $z$ exists in $[\alpha ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \beta ]$, it satisfies $\varPhi (\alpha ) = \alpha \ge 0$ and $\varPhi (\beta ) = s\beta - (1 - s)(k - 1)\bar{\varepsilon }$, then we have,

• (1) As $\varPhi (\beta ) = s\beta - (1 - s)(k - 1)\bar{\varepsilon } \le 0$, there is a unique optimal $z_{0} \in [\alpha ,{\kern 1pt} {\kern 1pt} {\kern 1pt} \beta ]$ satisfies $\varPhi (z_{0} ) = 0$.

• (2) As $\varPhi (\beta ) = s\beta - (1 - s)(k - 1)\bar{\varepsilon } > 0$, then there should be $z \in (\alpha ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \beta ]$ satisfies $\varPhi (z) > 0$. Combining $y_{0}^{{1/k}} (nq\mu )^{{1 - 1/k}} /(z^{{2 - 1/k}} k) > 0$, we know $\varPi_{DE} (z,w){\kern 1pt}$ is an increasing function in $z \in [\alpha ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \beta ]$. Obviously, the maximum $\varPi_{DE} (z,w){\kern 1pt}$ can be obtained as $z_{0} = \beta$.□

Proof for Theorem 3

From Theorem 2 we have, the optimal stocking factor is not impacted by production quantity $q$. Taking the second partial derivatives of $\varPi_{DE} (q,w(q))$ with respect to $q$, we have

$$ {{d^{2} \varPi_{DE} (q,w(q)){\kern 1pt} } \mathord{\left/ {\vphantom {{d^{2} \varPi_{DE} (q,w(q)){\kern 1pt} } {dq^{2} }}} \right. \kern-0pt} {dq^{2} }} = - \frac{1}{k}A_{0} E_{\mu } [\mu^{{1 - {1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0pt} k}}} ] \cdot n^{{1 - {1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0pt} k}}} q^{{ - {1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0pt} k} - 1}} - 4c_{2} \cdot {{n\bar{\mu }} \mathord{\left/ {\vphantom {{n\bar{\mu }} {\kappa_{1} (\eta )}}} \right. \kern-0pt} {\kappa_{1} (\eta )}} < 0 $$

Therefore, $\varPi_{DE} (q,w(q))$ is a strictly concave function of $q$. Let ${{d\varPi_{DE} (q)} \mathord{\left/ {\vphantom {{d\varPi_{DE} (q)} {dq}}} \right. \kern-0pt} {dq}} = 0$, we can get the optimal production quantity $q_{DF}^{*}$

$$ {{d\varPi_{DE} (q,w(q)){\kern 1pt} } \mathord{\left/ {\vphantom {{d\varPi_{DE} (q,w(q)){\kern 1pt} } {dq}}} \right. \kern-0pt} {dq}} = A_{0} E_{\mu } [\mu^{{1 - {1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0pt} k}}} ] \cdot n^{{1 - {1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0pt} k}}} q^{{ - {1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0pt} k}}} - (c_{1} + 4c_{2} q) \cdot n{{\bar{\mu }} \mathord{\left/ {\vphantom {{\bar{\mu }} {\kappa_{1} (\eta )}}} \right. \kern-0pt} {\kappa_{1} (\eta )}} = 0 $$

From (6), we get the optimal purchasing price $w_{DE}^{*} = {{(c_{1} + 2c_{2} q_{DF}^{*} )} \mathord{\left/ {\vphantom {{(c_{1} + 2c_{2} q_{DF}^{*} )} {\kappa_{1} (\eta )}}} \right. \kern-0pt} {\kappa_{1} (\eta )}}$.□

Proof for Theorem 4

Similar to the proof of Theorem 2.□

Proof for Theorems 5 and 6

Similar to the proof of Theorem 3.□

Proof for Lemma 1

From Theorem 3, taking the first order partial derivatives of $q_{DF}^{*}$ with respect to $\eta$, we have

$$ \frac{{\partial q_{DF}^{*} }}{\partial \eta } = \kappa_{1} (\eta )^{ - 1} \frac{{q_{DF}^{*} (c_{1} + 4c_{2} q_{DF}^{*} )k}}{{c_{1} + 4(k + 1)c_{2} q_{DF}^{*} }} \cdot \frac{{\partial \kappa_{1} (\eta )}}{\partial \eta } $$

In addition, we know that

$$ \frac{{\partial \kappa_{1} (\eta )}}{\partial \eta } = \frac{1}{{\eta^{2} }}\int_{A}^{{G^{ - 1} (\mu )}} {G(\mu )d\mu } > 0 $$

Therefore, ${{\partial q_{DF}^{*} } \mathord{\left/ {\vphantom {{\partial q_{DF}^{*} } {\partial \eta }}} \right. \kern-0pt} {\partial \eta }} > 0$.

Taking the first order partial derivatives of $w_{DE}^{*}$ with respect to $\eta$, we have

$$ \kappa_{1} (\eta )^{2} \frac{{\partial w_{DE}^{*} }}{\partial \eta } = 2c_{2} \kappa_{1} (\eta ))\frac{{\partial q_{DF}^{*} }}{\partial \eta } - \frac{{\partial \kappa_{1} (\eta )}}{\partial \eta }(c_{1} + 2c_{2} q_{DF}^{*} ) $$

Hence,

$$ \frac{{\partial w_{DE}^{*} }}{\partial \eta } = - \frac{{(c_{1} + 2c_{2} q_{DF}^{*} )(c_{1} + 4c_{2} q_{DF}^{*} ) + 2kc_{1} c_{2} q_{DF}^{*} }}{{[c_{1} + 4(k + 1)c_{2} q_{DF}^{*} ] \cdot \kappa_{1} (\eta )^{2} }} \cdot \frac{{\partial \kappa_{1} (\eta )}}{\partial \eta } < 0 $$

where $\frac{{\partial \kappa_{1} (\eta )}}{\partial \eta } = \frac{1}{{\eta^{2} }}\int_{A}^{{G^{ - 1} (\mu )}} {G(\mu )d\mu } > 0$.Since $p_{DE}^{*} (w) = {{[y_{0} z_{0} } \mathord{\left/ {\vphantom {{[y_{0} z_{0} } {(nq\mu )}}} \right. \kern-0pt} {(nq\mu )}}]^{{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0pt} k}}}$, we can easily obtain ${{dp_{DE}^{*} } \mathord{\left/ {\vphantom {{dp_{DE}^{*} } {d\eta }}} \right. \kern-0pt} {d\eta }} < 0$.□

Proof or Lemma 2

• ① It can be easily seen from Theorem 2.

• ② It can be easily seen from Theorem 3 that $q_{DF}^{*}$ is not related to $c_{0}$. In addition, from (15), we have ${{\partial q_{DF}^{*} } \mathord{\left/ {\vphantom {{\partial q_{DF}^{*} } {\partial c_{1} = }}} \right. \kern-0pt} {\partial c_{1} = }}{{\bar{\mu }} \mathord{\left/ {\vphantom {{\bar{\mu }} {[\kappa_{1} (\eta ) \cdot ({{d^{2} \varPi_{DE} (q,w(q)){\kern 1pt} } \mathord{\left/ {\vphantom {{d^{2} \varPi_{DE} (q,w(q)){\kern 1pt} } {d^{2} }}} \right. \kern-0pt} {d^{2} }}q)]}}} \right. \kern-0pt} {[\kappa_{1} (\eta ) \cdot ({{d^{2} \varPi_{DE} (q,w(q)){\kern 1pt} } \mathord{\left/ {\vphantom {{d^{2} \varPi_{DE} (q,w(q)){\kern 1pt} } {d^{2} }}} \right. \kern-0pt} {d^{2} }}q)]}}$, since ${{d^{2} \varPi_{DE} (q,w(q)){\kern 1pt} } \mathord{\left/ {\vphantom {{d^{2} \varPi_{DE} (q,w(q)){\kern 1pt} } {dq^{2} }}} \right. \kern-0pt} {dq^{2} }} < 0$, then ${{\partial q_{DF}^{*} } \mathord{\left/ {\vphantom {{\partial q_{DF}^{*} } {dc_{1} < 0}}} \right. \kern-0pt} {dc_{1} < 0}}$. Similarly, we can get that ${{\partial q_{DF}^{*} } \mathord{\left/ {\vphantom {{\partial q_{DF}^{*} } {\partial c_{2} = }}} \right. \kern-0pt} {\partial c_{2} = }}{{4\bar{\mu }q_{DF}^{*} } \mathord{\left/ {\vphantom {{4\bar{\mu }q_{DF}^{*} } {[\kappa_{1} (\mu ) \cdot {{d^{2} \varPi_{DE} (q,w(q)){\kern 1pt} } \mathord{\left/ {\vphantom {{d^{2} \varPi_{DE} (q,w(q)){\kern 1pt} } {dq^{2} }}} \right. \kern-0pt} {dq^{2} }}]}}} \right. \kern-0pt} {[\kappa_{1} (\mu ) \cdot {{d^{2} \varPi_{DE} (q,w(q)){\kern 1pt} } \mathord{\left/ {\vphantom {{d^{2} \varPi_{DE} (q,w(q)){\kern 1pt} } {dq^{2} }}} \right. \kern-0pt} {dq^{2} }}]}} < 0$.

• ③ Since $p_{DE}^{*} (w) = {{[y_{0} z_{0} } \mathord{\left/ {\vphantom {{[y_{0} z_{0} } {(q\mu )}}} \right. \kern-0pt} {(q\mu )}}]^{{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0pt} k}}}$, from (1) and (2), we can get (3).

• ④ It can be easily seen from Theorem 3 that $w_{DE}^{*}$ is not related to $c_{0}$. From Theorem 3, we can obtain $\kappa_{1} (\eta )\frac{{\partial w_{DE}^{*} }}{{\partial c_{1} }} = 1 + 2c_{2} \frac{{\partial q_{DF}^{*} }}{{\partial c_{1} }} = 1 - \frac{{2kc_{2} }}{{(c_{1} + 4c_{2} q_{DF}^{*} ) \cdot q_{DF}^{* - 1} + 4kc_{2} }} > 0$. Similarly, we have $\kappa (\mu ,\eta )\frac{{\partial w_{DE}^{*} }}{{\partial c_{2} }} = 2q_{DF}^{*} + 2c_{2} \frac{{\partial q_{DF}^{*} }}{{\partial c_{2} }} = 2q_{DF}^{*} \left\{ {1 - \frac{{4kc_{2} }}{{(c_{1} + 4c_{2} q_{DF}^{*} ) \cdot q_{DF}^{* - 1} + 4kc_{2} }}} \right\} > 0$. Hence, $\frac{{\partial w_{DE}^{*} }}{{\partial c_{1} }} > 0$ and $\frac{{\partial w_{DE}^{*} }}{{\partial c_{2} }} > 0$.□

Proof for Lemma 3

We use Reduction to Absurdity to prove Lemma 3. Assume $q_{DF}^{*} \ge q_{SC}^{*}$ first. Since $c_{1} > 0,c_{2} > 0$ and $k > 1$, then $q_{DF}^{*} [c_{1} + 4c_{2} q_{DF}^{*} ]^{k} > q_{SC}^{*} [c_{1} + 2c_{2} q_{SC}^{*} ]^{k}$, which is not consistent with the satisfaction condition $nq_{DF}^{*} [c_{1} + 4c_{2} q_{DF}^{*} ]^{k} < A_{0}^{k} E_{\mu } [\mu^{{1 - {1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0pt} k}}} ]^{k}$$= nq_{SC}^{*} [c_{1} + 2c_{2} q_{SC}^{*} ]^{k}$ for Theorems 2 and 3. Hence, we know that $q_{DF}^{*} < q_{SC}^{*}$.

Moreover, from Lemma 1 we know, $q_{DF}^{*}$ is decreasing in $\eta$. Hence, $q_{DF}^{*}$ is smaller for the more risk-averse farmer, thus the difference of the optimal production quantity under centralized and decentralized decision models, $(q_{SC}^{*} - q_{DF}^{*} )$, is larger.

Since the optimal retail price is negative correlation with the optimal production quantity, hence, $\left( {p_{SC}^{*} - p_{DE}^{*} } \right)$ decreases with decreasing $\eta$.□

Appendix 2

According to the general definition of CVaR, by taking the market demand uncertainty into consideration, the farmer’s decision making function can be expressed by

$$ \begin{aligned} CVaR_{\eta } (\hbox{min} (z,x) + s(z - x)^{ + } ) & = v_{\varepsilon } - \frac{1}{\eta }E[v_{\varepsilon } - \hbox{min} (z,x) - s(z - x)^{ + } ]^{ + } \\ {\kern 1pt} & = v_{\varepsilon } - \frac{1}{\eta }\int_{\alpha }^{z} {\left( {v_{\varepsilon } - sz - (1 - s)x} \right)^{ + } dF(x)} - \frac{1}{\eta }\int_{z}^{\beta } {(v_{\varepsilon } - z)^{ + } dF(x)} \\ \end{aligned} $$

where $[Z]^{ + } = \hbox{max} [Z,0]$.

For any given $z$, it can be simplified in the following three cases for solving ${\kern 1pt} \mathop {\hbox{max} }\limits_{{v_{\varepsilon } \in R}} {\text{C}}VaR_{\eta } (z,v_{\varepsilon } )$.

• (1) If $v_{\varepsilon } \le sz + (1 - s)\alpha$, $CVaR_{\eta } (z,v_{\varepsilon } ) = v_{\varepsilon }$, then ${{\partial CVaR_{\eta } (z,v_{\varepsilon } )} \mathord{\left/ {\vphantom {{\partial CVaR_{\eta } (z,v_{\varepsilon } )} {\partial v_{\varepsilon } }}} \right. \kern-0pt} {\partial v_{\varepsilon } }} = 1 > 0$.

• (2) If $sz + (1 - s)\alpha < v_{\varepsilon } \le z$, ${\kern 1pt} CVaR_{\eta } (z,v_{\varepsilon } ) = v_{\varepsilon } - \frac{1}{\eta }\int_{\alpha }^{{\frac{{v_{\varepsilon } - sz}}{1 - s}}} {\left( {v_{\varepsilon } - sz - (1 - s)x} \right)dF(x)}$, then

  $$ \frac{{\partial CVaR_{\eta } (z,v_{\varepsilon } )}}{{\partial v_{\varepsilon } }} = 1 - \frac{1}{\eta }F\left( {\frac{{v_{\varepsilon } - sz}}{1 - s}} \right) $$

  Notice that

  $$ \frac{{\partial CVaR_{\eta } (z,v_{\varepsilon } )}}{{\partial v_{\varepsilon } }}\left| {_{{v_{\varepsilon } = sz + (1 - s)\alpha }} = 1 > 0} \right.,\quad \frac{{\partial CVaR_{\eta } (z,v_{\varepsilon } )}}{{\partial v_{\varepsilon } }}\left| {_{{v_{\varepsilon } = z}} = } \right.1 - \frac{1}{\eta }F(z) $$

  Hence, as $1 - \frac{1}{\eta }F(z) > 0 \Rightarrow F(z) < \eta$, ${{\partial CVaR_{\eta } (z,v_{\varepsilon } )} \mathord{\left/ {\vphantom {{\partial CVaR_{\eta } (z,v_{\varepsilon } )} {\partial v_{\varepsilon } }}} \right. \kern-0pt} {\partial v_{\varepsilon } }}\left| {_{{v_{\varepsilon } = z}} } \right. > 0$, we have $v_{\varepsilon }^{*} = z$. Otherwise, as $1 - \frac{1}{\eta }F(z) \le 0 \Rightarrow F(z) \ge \eta$, we have $v_{\varepsilon }^{*} = sz + (1 - s)F^{ - 1} (\eta )$.

• (3) If $v_{\varepsilon } > z$, $CVaR_{\eta } (z,v_{\varepsilon } ) = v_{\varepsilon } - \frac{1}{\eta }\int_{\alpha }^{z} {\left( {v_{\varepsilon } - sz - (1 - s)x} \right)dF(x)} - \frac{1}{\eta }\int_{z}^{\beta } {(v_{\varepsilon } - z)dF(x)}$, then

  $$ \frac{{\partial CVaR_{\eta } (z,v_{\varepsilon } )}}{{\partial v_{\varepsilon } }} = 1 - \frac{1}{\eta } < 0 $$

Hence,

• ① If $\eta \le F(z)$, the optimal value of $v_{\varepsilon }^{*}$ satisfies $v_{\varepsilon }^{*} = sz + (1 - s)F^{ - 1} (\eta )$, then

  $$ {\kern 1pt} CVaR_{\eta } (z,v_{\varepsilon } ) = F^{ - 1} (\eta ) - \frac{1}{\eta }\int_{\alpha }^{{F^{ - 1} (\eta )}} {\left( {F^{ - 1} (\eta ) - sz - (1 - s)x} \right)dF(x)} = sz + (1 - s)\frac{1}{\eta }\int_{\alpha }^{{F^{ - 1} (\eta )}} {xdF(x)} $$

  Then we have

  $$ {\kern 1pt} \frac{{dCVaR_{\eta } (z,v_{\varepsilon } )}}{dz} = s \ge 0 $$

  $CVaR_{\eta } (z,v_{\varepsilon } )$ is an increasing function of $z$, there is no extreme point.

• ② If $F(z) < \eta$, the optimal value of $v_{\varepsilon }^{*}$ satisfies $v_{\varepsilon }^{*} = z$, then

  $$ {\kern 1pt} CVaR_{\eta } (z,v_{\varepsilon } ) = z - (1 - s)\frac{1}{\eta }\int_{\alpha }^{z} {\left( {z - x} \right)dF(x)} . $$

  □

Appendix 3

According to the general definition of CVaR, by taking the yield uncertainty into consideration, the farmer’s decision making function can be expressed by

$$ CVaR_{\eta } (\mu^{k} ,v_{\mu } ) = {\kern 1pt} v_{\mu } - \frac{1}{\eta }E(v_{\mu } - \mu^{k} )^{ + } {\kern 1pt} {\kern 1pt} {\kern 1pt} = v_{\mu } - \frac{1}{\eta }\int_{A}^{B} {(v_{\mu } - \mu^{k} )^{ + } dG(\mu )} $$

where $k > 0$ is an any real number.

Similar to the solve process for Appendix B, we have

$$ CVaR_{\eta } (\mu^{k} ,v_{\mu } ) = \frac{1}{\eta }\int_{A}^{{G^{ - 1} (\eta )}} {\mu^{k} } dG(\mu ). $$

□