Joint optimal ordering and weather hedging decisions: mean-CVaR model

Abstract

This paper considers the problem of hedging inventory risk for a seasonal product whose demand is sensitive to weather conditions, such as the average seasonal temperature. The newsvendor not only decides the order quantity, but also adopts a weather hedging strategy. A typical hedging strategy is to use an option (weather derivative) that is constructed on a weather index before the season begins, which will compensate the buyer of the option if the actual seasonal weather index is above (or below) a given strike level. We adopt the risk measure of Conditional-Value at Risk ($\hbox{CVaR}$) and explore the joint decision problem in mean-CVaR criterion. We find that the weather derivative hedging can increase order quantity. Furthermore, it can help the risk-averse newsvendor improve both expected overall and downside profits.

Appendix

Proof of Lemma 1

For any $\lambda \in[0,1]$, and any two different points $(Q_1,n_1,v_1)$ and $(Q_2,n_2,v_2)$,

$$ \begin{aligned} F_{\beta}(\lambda(Q_1,n_1,v_1)+(1-\lambda)(Q_2,n_2,v_2)) &=F_{\beta}(\lambda Q_1+(1-\lambda)Q_2, \lambda n_1+(1-\lambda) n_2, \lambda v_1+(1-\lambda) v_2)\\ &= \lambda v_1+(1-\lambda)v_2-(1-\beta)^{-1}\hbox{E}[\lambda v_1+(1-\lambda) v_2-\Uppi(\lambda Q_1+(1-\lambda) Q_2, \lambda n_1+(1-\lambda) n_2, D(t))]^{+}. \end{aligned} $$

As $\Uppi(Q,n,D(t))$ is jointly concave in (Q, n),

$$ \begin{aligned} \Uppi(\lambda Q_1+(1-\lambda) Q_2, \lambda n_1+(1-\lambda) n_2, D(t)) =&\Uppi(\lambda(Q_1,n_1)+(1-\lambda)(Q_2,n_2), D(t))\\ &\ge \lambda \Uppi(Q_1,n_1,D(t))+(1-\lambda) \Uppi(Q_2,n_2,D(t)). \end{aligned} $$

Hence,

$$ \begin{aligned} [\lambda v_1+(1-\lambda) v_2-\Uppi(\lambda Q_1+(1-\lambda) Q_2, \lambda n_1+(1-\lambda) n_2, D(t))]^{+} &\le [\lambda v_1+(1-\lambda)v_2-\lambda \Uppi(Q_1,n_1,D(t))-(1-\lambda) \Uppi(Q_2,n_2,D(t))]^{+}\\ &= [\lambda(v_1-\Uppi(Q_1,n_1,D(t)))+(1-\lambda)(v_2-\Uppi(Q_2,n_2,D(t)))]^{+} \le \lambda[v_1-\Uppi(Q_1,n_1,D(t))]^{+}+(1-\lambda)[v_2-\Uppi(Q_2,n_2,D(t))]^{+}, \end{aligned} $$

the last inequality holds due to the convexity of function $[\cdot]^{+}$.

Then, substituting the above inequality into (13) yields

$$ \begin{aligned} F_{\beta}(\lambda(Q_1,n_1,v_1)+(1-\lambda)(Q_2,n_2,v_2)) &\ge \lambda v_1+(1-\lambda)v_2-(1-\beta)^{-1}\hbox{E}\{\lambda[v_1-\Uppi(Q_1,n_1,D(t))]^{+}+(1-\lambda)[v_2-\Uppi(Q_2,n_2,D(t))]^{+}\}\\ &= \lambda \{v_1-(1-\beta)^{-1}\hbox{E}[v_1-\Uppi(Q_1,n_1,D(t))]^{+}\}+(1-\lambda)\{v_2-(1-\beta)^{-1}\hbox{E}[v_2-\Uppi(Q_2,n_2,D(t))]^{+}\}\\ & =\lambda F_{\beta}(Q_1,n_1,v_1)+(1-\lambda) F_{\beta}(Q_2,n_2,v_2). \end{aligned} $$

Thus, F _β(Q, n, v) is jointly concave in (Q, n, v). $\square$

Proof of Proposition 1

Note the objective in (11) for the case without option hedging,

$$ \begin{aligned} \max_Q \{\lambda {\hbox{E}}[\Uppi_1(Q, D(t))] + (1-\lambda) \hat{\phi}_\beta(Q)\} &=\max_{Q}\{\lambda\hbox{E}[\Uppi_1(Q, D(t))]+(1-\lambda)\max_{v}\hat{F}_{\beta}(Q, v)\}\\ &=\max_{Q}\{\max_{v}\{\lambda\hbox{E}[\Uppi_1(Q, D(t))]+(1-\lambda)\hat{F}_{\beta}(Q, v)\}\}\\ &=\max_{(Q,v)}\{\lambda\hbox{E}[\Uppi_1(Q, D(t))]+(1-\lambda)\hat{F}_{\beta}(Q, v)\}. \end{aligned} $$

Let $\hat{F}_{(\beta, \lambda)}(Q, v)=\lambda\hbox{E}[\Uppi_1(Q, D(t))]+(1-\lambda)\hat{F}_{\beta}(Q, v)=\lambda\hbox{E}[\Uppi_1(Q, D(t))]+(1-\lambda)\{v-(1-\beta)^{-1}\hbox{E}[v-\Uppi_1(Q, D(t))]^{+}\}.$ Denote by D ^l(t) and D ^u(t) the lower and upper bounds of D(t), respectively. For fixed Q,

$$ \begin{aligned} \frac{\partial\hat{ F}_{(\beta, \lambda)}(Q, v)}{\partial v} &=(1-\lambda)\left\{1-{\frac{1}{1-\beta}}\hbox{E}[1_{v>\Uppi_1(Q, D(t))}]\right\} \\&=(1-\lambda)\left\{1-{\frac{1}{1-\beta}}\hbox{E}_{t}\left[\int\limits_{D^{l}(t)}^{Q}1_{v>(p-s)x-(c-s)Q}\cdot g(x;t)dx +\int\limits_{Q}^{D^{u}(t)}1_{v>(p-c)Q}\cdot g(x;t)dx\right ]\right\} \\ &=(1-\lambda)\left\{1-{\frac{1}{1-\beta}}\hbox{E}_{t} \left[\int\limits_{D^{l}(t)}^{\min(Q,{\frac{v+(c-s)Q} {p-s}})}g(x;t)dx +\int\limits_{Q}^{D^{u}(t)}1_{v>(p-c)Q}\cdot g(x;t)dx\right ]\right\}\\ &=\left\{ \begin{array}{ll} (1-\lambda)\left\{1-{\frac{1} {1-\beta}} \hbox{E}_{t}\left[\int\limits_{D^{l}(t)}^{{\frac{v+(c-s)Q} {p-s}}}g(x;t)dx\right ] \right\} &\hbox{if}\,v\le(p-c)Q,\\ (1-\lambda) \left[-{\frac{\beta}{1-\beta}}\right ] &\hbox{if}\,v>(p-c)Q. \end{array} \right . \end{aligned} $$

At the point (p − c)Q,

$$ \begin{aligned} {\frac{\partial^{-} \hat{F}_{(\beta, \lambda)}(Q,v)}{\partial v}}|_{v=(p-c)Q}&=(1-\lambda)\left\{1-{\frac{1} {1-\beta}}\hbox{E}_{t}\left[\int\limits_{D^{l}(t)}^{Q}g(x;t)dx\right]\right \} ,\\ {\frac{\partial^{+} \hat{F}_{(\beta, \lambda)}(Q,v)}{\partial v}}|_{v=(p-c)Q}&=(1-\lambda)\left[-{\frac{\beta}{1-\beta}}\right] \le 0. \end{aligned} $$

• (1) If ${\frac{\partial^{-} \hat{F}_{(\beta, \lambda)}(Q,v)} {\partial v}}|_{v=(p-c)Q}<0$, the maximizer $\hat{v}$ of $\hat{F}_{(\beta, \lambda)}(Q,v)$ satisfies $\hat{v}<(p-c)Q$ and is determined by

  $$ 1-{\frac{1} {1-\beta}}\hbox{E}_{t}\left[\int\limits_{D^{l}(t)}^{{\frac{\hat{v}+(c-s)Q} {p-s}}}g(x;t)dx\right] =0. $$

  (13)

  In this case,

  $$\begin{aligned} \hat{F}_{(\beta, \lambda)}(Q,\hat{v})= & \lambda\hbox{E}[\pi_1(Q,D(t))] &+(1-\lambda)\left\{\hat{v}-(1-\beta)^{-1}\hbox{E}_{t}\left[\int\limits_{D^{l}(t)}^{{\frac{\hat{v}+(c-s)Q} {p-s}}}(\hat{v} +(c-s)Q-(p-s)x)g(x;t)dx\right]\right \} \\ {\frac{d\hat{F}_{(\beta, \lambda)}(Q, \hat{v})}{d Q}} &= {\frac{\partial \hat{F}_{(\beta, \lambda)}(Q, \hat{v})} {\partial Q}}+{\frac{\partial \hat{F}_{(\beta, \lambda)}(Q, \hat{v})} {\partial v}} \frac{\partial \hat{v}}{\partial Q}\\ &=\lambda\{-c+s+(p-s)\hbox{E}[1_{D(t)> Q}]\} +(1-\lambda) \left\{(1-\beta)^{-1}(-c+s)\hbox{E}_{t} \left[\int\limits_{D^{l}(t)}^{{\frac{\hat{v}+(c-s)Q} {p-s}}}g(x;t)dx\right]\right\}+0\\ &=\lambda\{-c+s+(p-s)\hbox{E}[1_{D(t)> Q}]\}+(1-\lambda)(-c+s)\\ &=\lambda\{-c+s+(p-s)(1-\hbox{E}_{t}[G(Q; t)])\}+(1-\lambda)(-c+s)\\ &=\lambda(p-c)-\lambda(p-s)\hbox{E}_{t}[G(Q; t)]-(1-\lambda)(c-s). \end{aligned}$$

  Then $\hat{Q}$ satisfies $\hbox{E}_{t}[G(\hat{Q}; t)]={\frac{\lambda (p-c)-(1-\lambda)(c-s)}{\lambda (p-s)}}=1-{\frac{c-s}{\lambda (p-s)}}$, where $\lambda >{\frac{c-s}{\beta (p-s)}}$ to make the condition ${\frac{\partial^{-} \hat{F}_{(\beta, \lambda)}(\hat{Q},v)}{\partial v}}|_{v=(p-c)\hat{Q}}<0$ hold true.

• (2) If ${\frac{\partial^{-} \hat{F}_{(\beta, \lambda)}(Q,v)}{\partial v}}|_{v=(p-c)Q}\ge 0$, note that the auxiliary function is concave in v, then $\hat{v}=(p-c)Q$. In this case,

  $$ \hat{F}_{(\beta, \lambda)}(Q, (p-c)Q)=\lambda\hbox{E}[\Uppi_1(Q, D(t))] +(1-\lambda)\left\{(p-c)Q-(1-\beta)^{-1}\hbox{E}_{t}\left[\int\limits_{D^{l}(t)}^{Q}((p-s)Q -(p-s)x)g(x;t)dx\right]\right \} , $$

  then

  $$ \begin{aligned}{\frac{d\hat{F}_{(\beta,\lambda)}(Q, (p-c)Q)}{d Q}}&=\lambda \{-c+s+(p-s)\hbox{E}[1_{D(t)> Q}]\} +(1-\lambda)\left\{(p-c)-(1-\beta)^{-1}(p-s)\hbox{E}_{t}\left[\int\limits_{D^{l}(t)}^{Q}g(x; t)dx\right]\right\} \\ &= \lambda\{-c+s+(p-s)(1-\hbox{E}_{t}[G(Q; t)])\} +(1-\lambda)\{(p-c)-(1-\beta)^{-1}(p-s)\hbox{E}_{t}[G(Q; t)]\}\\ &=(p-c)-(\lambda +(1-\lambda)(1-\beta)^{-1})(p-s)\hbox{E}_{t}[G(Q; t)]. \end{aligned} $$

  Then $\hat{Q}$ satisfies $\hbox{E}_{t}[G(\hat{Q}; t)]={\frac{(1-\beta)(p-c)}{(1-\lambda\beta)(p-s)}}$, where $\lambda \le {\frac{c-s}{\beta (p-s)}}$ to make the condition ${\frac{\partial^{-} \hat{F}_{(\beta, \lambda)}(\hat{Q},v)} {\partial v}}|_{v=(p-c)\hat{Q}}\ge 0$ hold true.

From the above analysis, the optimal order quantity without option hedging under the tradeoff criterion, $\hat{Q}$, is determined by (12). $\square$

We give here Lemma 2 which will be repeatedly used in the proofs of the results to follow.

Lemma 2

Let$G_{1}(\cdot)$and$G_{2}(\cdot)$be two integrable functions. If they have contrary monotonicity, then$\hbox{E}[G_1(t)G_2(t)]\le\hbox{E}[G_1(t)]\hbox{E}[G_2(t)]$, and if they have same monotonicity, then$\hbox{E}[G_1(t)G_2(t)]\ge\hbox{E}[G_1(t)]\hbox{E}[G_2(t)],$wheretis a continuous random variable.

Proof

If $G_1(\cdot)$ and $G_2(\cdot)$ have contrary monotonicity, for any realizations t and s, we have $[G_1(t)-G_1(s)][G_2(t)-G_2(s)]\le 0 \Rightarrow G_1(t) G_2(t)+G_1(s) G_2(s) \le G_1(t) G_2(s)+G_1(s) G_2(t)$. Assuming the density function of t to be $f(\cdot),$

$$ \begin{aligned} \hbox{E}[G_1(t)]\hbox{E}[G_2(t)] :&= \hbox{E}[G_1(t)]\hbox{E}[G_2(s)]\\ &=\int\limits_{t}G_1(t)f(t)dt \int\limits_{s}G_2(s)f(s)ds\\ &=\int\limits_{t}G_1(t)f(t)\left[\int\limits_{s}G_2(s)f(s)ds\right]dt\\ &=\int\int\limits_{t\times s} G_1(t) G_2(s) f(t) f(s) ds dt\\ &=\int\int\limits_{t\times s}{\frac{G_1(t) G_2(s)+G_1(s) G_2(t)}{2}}f(t) f(s) ds dt\\ &\ge\int\int\limits_{t\times s}{\frac{G_1(t) G_2(t)+G_1(s) G_2(s)}{2}}f(t) f(s) ds dt\\ &=\int\limits_{t}G_{1}(t)G_{2}(t)f(t)dt\\ &=\hbox{E}[G_1(t)G_2(t)], \end{aligned} $$

i.e., $\hbox{E}[G_1(t)G_2(t)]\le\hbox{E}[G_1(t)]\hbox{E}[G_2(t)].$ Similarly, if $G_1(\cdot)$ and $G_2(\cdot)$ have same monotonicity, it holds that $\hbox{E}[G_1(t)G_2(t)]\ge\hbox{E}[G_1(t)]\hbox{E}[G_2(t)]$. $\square$

Proof of Proposition 4

Note the objective in (7) for the case with option hedging,

$$ \begin{aligned} \max_{(Q,n)} \{\lambda {\hbox{E}}[\Uppi(Q, n, D(t))] + (1-\lambda) \phi_\beta(Q,n)\}&= \max_{(Q,n)}\{\lambda\hbox{E}[\Uppi(Q,n, D(t))]+(1-\lambda)\max_{v}F_{\beta}(Q,n,v)\}\\ &= \max_{(Q,n,v)}\{\lambda\hbox{E}[\Uppi(Q,n, D(t))]+(1-\lambda)F_{\beta}(Q,n,v)\}. \end{aligned} $$

Let $F_{(\beta, \lambda)}(Q,n,v)=\lambda\hbox{E}[\Uppi(Q,n,D(t))]+(1-\lambda)F_{\beta}(Q,n,v)=\lambda\hbox{E}[\Uppi(Q,n, D(t))]+(1-\lambda)\{v-(1-\beta)^{-1}\hbox{E}[v-\Uppi(Q,n,D(t))]^{+}\}.$

$$ {\frac{\partial F_{(\beta,\lambda)}(Q,n,v)}{\partial n}}|_{n=0}=\lambda\hbox{E}[\Uppi_2(t)]+{\frac{1-\lambda} {1-\beta}}\hbox{E}[\Uppi_2(t)\cdot1_{\Uppi_1(Q,D(t))<v}]. $$

Both $\Uppi_2(t)$ and $\hbox{E}_{D}[1_{\Uppi_1(Q,D(t))<v}|t]$ are increasing in t (It is easy to check that for given $Q, \Uppi_1(Q, \cdot)$ is an increasing function and $1_{\Uppi_1(Q, \cdot)<v}$ is a decreasing function then $\hbox{E}_{D}[1_{\Uppi_1(Q,D(t))<v}|t]$ is increasing in t by the assumption that D(t) is stochastically decreasing in t.), so by Lemma 2, if $\hbox{E}[\Uppi_2(t)]=0$,

$$ {\frac{\partial F_{(\beta,\lambda)}(Q,n,v)}{\partial n}}|_{n=0}\ge {\frac{1-\lambda} {1-\beta}}\hbox{E}[\Uppi_2(t)]\hbox{E}[1_{\Uppi_1(Q,D(t))<v}]=0. $$

By concavity of $F_{(\beta, \lambda)}(Q,n,v), n^{*}\ge 0$. $\square$

Proof of Proposition 5

$$ \begin{aligned} {\frac{\partial F_{(\beta,\lambda)}(Q,n,v)}{\partial v}}&=(1-\lambda)\{1-(1-\beta)^{-1}\hbox{E}[1_{v>\Uppi(Q, n, D(t))}]\}\\ &= (1-\lambda)\left\{1-(1-\beta)^{-1}\hbox{E}_{t}\left[\int\limits_{D^l(t)}^{Q}1_{v>(p-s)x-(c-s)Q+n\Uppi_2(t)}\cdot g(x; t) dx +\int\limits_{Q}^{D^u(t)}1_{v>(p-c)Q+n\Uppi_2(t)}\cdot g(x; t)dx\right]\right \} \\ &=(1-\lambda)\left\{1-(1-\beta)^{-1}\hbox{E}_{t}\left[\int\limits_{D^{l}(t)}^{\min(Q,{\frac{v+(c-s)Q-n\Uppi_2(t)} {p-s}})}g(x; t)dx +\int\limits_{Q}^{D^{u}(t)}1_{v>(p-c)Q+n\Uppi_2(t)}\cdot g(x; t)dx\right]\right \} . \end{aligned} $$

(14)

If $n>0, {\frac{\partial F_{(\beta, \lambda)}(Q, n, v)}{\partial v}}|_{v=(-c+s)Q+n\Uppi_2^l}=1-\lambda,$ and

$$ \begin{aligned} {\frac{\partial ^{-}F_{(\beta, \lambda)}(Q, n, v)}{\partial v}}|_{v=(p-c)Q+n\Uppi_2^u} &= (1-\lambda)\left\{1-(1-\beta)^{-1}\hbox{E}_{t}\left[\int\limits_{D^l(t)}^{Q}g(x; t)dx+\int\limits_{Q}^{D^u(t)} 1_{\Uppi_2^u>\Uppi_2(t)}\cdot g(x; t)dx\right]\right\} \\ &=(1-\lambda)\left\{1-(1-\beta)^{-1}\hbox{E}_{t}\left[G(Q; t)+1_{\Uppi_2^u>\Uppi_2(t)}(1-G(Q; t))\right ]\right\} , {\frac{\partial^{+} F_{(\beta, \lambda)}(Q, n, v)}{\partial v}}|_{v=(p-c)Q+n\Uppi_2^u}\\ &=(1-\lambda)\left\{1-(1-\beta)^{-1}\hbox{E}_{t}\left[\int\limits_{D^l(t)}^{Q}g(x; t)dx+\int\limits_{Q}^{D^u(t)}g(x; t)dx\right ]\right\}\\ &=(1-\lambda)(1-(1-\beta)^{-1})<0. \end{aligned} $$

From Proposition 1,

$$ \left\{ \begin{array}{ll} \hbox{E}_{t}[G(\hat{Q}; t)]=1-{\frac{c-s}{\lambda (p-s)}}>1-\beta,& \lambda > {\frac{c-s}{\beta (p-s)}},\\ \hbox{E}_{t}[G(\hat{Q}; t)]={\frac{(1-\beta)(p-c)}{(1-\lambda \beta)(p-s)}}\le 1-\beta, &\lambda \le {\frac{c-s}{\beta (p-s)}}. \end{array} \right . $$

• (1) If $\lambda > {\frac{c-s}{\beta (p-s)}}$, then

  $$ {\frac{\partial^{-} F_{(\beta,\lambda)}(\hat{Q}, n, v)}{\partial v}}|_{v=(p-c)\hat{Q}+n\Uppi_2^u}\le(1-\lambda)\{1-(1-\beta)^{-1}\hbox{E}_{t}[G(\hat{Q}; t)]\}< 0 $$

  by noticing that $\hbox{E}_{t}[1_{\Uppi_2^u>\Uppi_2(t)}(1-G(\hat{Q}; t))]\ge 0.$ Thus, $v^*(\hat{Q}, n)\in((-c+s)\hat{Q}+n\Uppi^l_2, (p-c)\hat{Q}+n\Uppi^u_2)$, and by (14),

  $$ \hbox{E}[1_{v^*(\hat{Q}, n)>\Uppi(\hat{Q},n,D(t))}]=1-\beta; $$

• (2) If $\lambda \le {\frac{c-s}{\beta (p-s)}}$, and if ${\frac{\partial^{-} F_{(\beta, \lambda)}(Q, n, v)}{\partial v}}|_{v=(p-c)\hat{Q}+n\Uppi_2^u}\ge 0$, then $v^*(\hat{Q}, n)=(p-c)\hat{Q}+n\Uppi^u_2$, and

  $$ \begin{aligned} F_{(\beta, \lambda)}(Q, n, v)|_{v=(p-c)Q+n\Uppi_2^u} &= \lambda\hbox{E}[\Uppi(Q,n,D(t))]+(1-\lambda)\{(p-c)Q+n\Uppi_2^u\\ &\quad-(1-\beta)^{-1}\hbox{E}_{t}\left[\int\limits_{D^l(t)}^{Q}((p-s)(Q-x)+n(\Uppi^u_2-\Uppi_2(t))) g(x; t) dx +\int\limits_{Q}^{D^u(t)}n(\Uppi_2^u-\Uppi_2(t))g(x; t)dx]\right \},\\ & {\frac{\partial F_{(\beta, \lambda)}(Q, n, v^*(Q, n))}{\partial Q}}|_{Q=\hat{Q}}&=\lambda \{p-c-(p-s)\hbox{E}_{t}[G(\hat{Q}; t)]\} \quad+(1-\lambda)\left\{(p-c)-(1-\beta)^{-1}(p-s)\hbox{E}_{t}\left[\int\limits_{D^l(t)}^{\hat{Q}}g(x; t)dx\right]\right\}\\ &= (p-c)-{\frac{1-\lambda\beta} {1-\beta}}(p-s)\hbox{E}_{t}[G(\hat{Q}; t)] =0; \end{aligned} $$

• (3) If $\lambda \le {\frac{c-s}{\beta (p-s)}}$, and ${\frac{\partial^{-} F_{(\beta, \lambda)}(Q, n, v)}{\partial v}}|_{v=(p-c)\hat{Q}+n\Uppi_2^u}<0$, then $v^*(\hat{Q}, n)\in((-c+s)\hat{Q}+n\Uppi^l_2, (p-c)\hat{Q}+n\Uppi^u_2)$, and

  $$ \hbox{E}[1_{v^*(\hat{Q}, n)>\Uppi(\hat{Q},n,D(t))}]=1-\beta. $$

For the cases (1) and (3), i.e., $\hbox{E}[1_{v^*(\hat{Q}, n)>\Uppi(\hat{Q},n,D(t))}]=1-\beta,$

$$ \begin{aligned} {\frac{\partial F_{(\beta, \lambda)}(Q,n,v)}{\partial Q}}|_{Q=\hat{Q}}&= \lambda\hbox{E}\left [{\frac{\partial \Uppi(\hat{Q},n,D(t))}{\partial Q}}\right ]+{\frac{1-\lambda} {1-\beta}}\hbox{E}\left [1_{v>\Uppi(\hat{Q},n,D(t))}{\frac{\partial \Uppi(\hat{Q},n,D(t))}{\partial Q}}\right ]\\ & =\lambda \{(p-c)-(p-s)\hbox{E}_{t}[G(\hat{Q};t)]\}\\ &\quad+{\frac{1-\lambda}{1-\beta}}\hbox{E}[1_{v>\Uppi(\hat{Q},n,D(t))}(p-c-(p-s)1_{D(t)< \hat{Q}})]\\ &=\lambda(p-c)-\lambda(p-s)\hbox{E}_{t}[G(\hat{Q}; t)]\\ &\quad+{\frac{1-\lambda}{1-\beta}}\{(p-c)\hbox{E}[1_{v>\Uppi(\hat{Q},n,D(t))}]-(p-s)\hbox{E}[1_{v>\Uppi(\hat{Q},n,D(t))} 1_{D(t)< \hat{Q}}]\}. \end{aligned} $$

When $\lambda >{\frac{c-s}{\beta (p-s)}}, \hbox{E}_{t}[G(\hat{Q}; t)]={\frac{\lambda (p-c)-(1-\lambda)(c-s)}{\lambda (p-s)}}$, and

$$ \begin{aligned} {\frac{\partial F_{(\beta, \lambda)}(Q,n,v^*(Q,n))}{\partial Q}}|_{Q=\hat{Q}} \ge&\lambda(p-c)-\lambda(p-s)\hbox{E}_{t}[G(\hat{Q}; t)] +{\frac{1-\lambda}{1-\beta}}\{(p-c)\hbox{E}[1_{v^*(\hat{Q},n)>\Uppi(\hat{Q},n,D(t))}]-(p-s)\hbox{E}[1_{v^*(\hat{Q},n)>\Uppi(\hat{Q},n,D(t))}]\}\\ &=(1-\lambda)(c-s)+{\frac{1-\lambda}{1-\beta}}\{(-c+s)\hbox{E}[1_{v^*(\hat{Q},n)>\Uppi(\hat{Q}, n, D(t))}]\}=0. \end{aligned} $$

When $\lambda \le {\frac{c-s}{\beta (p-s)}}, \hbox{E}_{t}[G(\hat{Q}; t)]={\frac{(1-\beta)(p-c)}{(1-\beta \lambda)(p-s)}}$, and

$$ \begin{aligned} {\frac{\partial F_{(\beta, \lambda)}(Q,n,v^*(Q,n))}{\partial Q}}|_{Q=\hat{Q}}\ge&\lambda(p-c)-\lambda(p-s)\hbox{E}_{t}[G(\hat{Q}; t)] +{\frac{1-\lambda}{1-\beta}}\{(p-c)\hbox{E}[1_{v^*(\hat{Q},n)>\Uppi(\hat{Q},n,D(t))}]-(p-s)\hbox{E}[1_{D(t)< \hat{Q}}]\}\\ &=\lambda(p-c)-{\frac{1-\beta \lambda}{1-\beta}}(p-s)\hbox{E}_{t}[G(\hat{Q}; t)]+{\frac{1-\lambda} {1-\beta}}(p-c)\hbox{E}[1_{v^*(\hat{Q},n)>\Uppi(\hat{Q},n,D(t))}]\\ &=(\lambda-1)(p-c)+{\frac{1-\lambda} {1-\beta}}(p-c)\hbox{E}[1_{v^*(\hat{Q},n)>\Uppi(\hat{Q},n,D(t))}] =0. \end{aligned} $$

In all, for all cases (1), (2) and (3), ${\frac{\partial F_{(\beta, \lambda)}(Q,n,v^*(Q,n))}{\partial Q}}|_{Q=\hat{Q}}\ge 0$.

When n < 0, similar analysis can be conducted by inter-changing the roles of $\Uppi_2^l$ and $\Uppi_2^u$ in the above proof, and when n = 0,

$$ {\frac{\partial F_{(\beta, \lambda)}(Q, 0, v^*(Q,0))}{\partial Q}}|_{Q=\hat{Q}}={\frac{\partial \hat{F}_{(\beta, \lambda)}(Q, \hat{v}(Q))}{\partial Q}}|_{Q=\hat{Q}}=0. $$

By concavity of F _{(β, λ)}(Q, n, v), we have $Q^*\ge \hat{Q}$. This completes the proof.

Note the condition in this proposition, G(y;t) is strictly increasing in y for given t, is to get a unique solution $\hat{Q}$ for the unhedged case. Generally without this condition, assume $\hat{Q}$ and Q ^* are the largest solutions for the unhedged and hedged cases respectively, we still have the result that $Q^*\ge \hat{Q}.$ $\square$