Buyback contract coordination in supply chain with fairness concern under demand updating: Enterprise Information Systems: Vol 15 , No 5

ABSTRACT

This paper sets supply chain buyback contract model with retailer's complete rationality and fairness concern respectively under stochastic demand, and uses Bayesian theorem to solve retailer's two-stage order decision under demand updating, so as to analyze the influence of retailer's fairness concern on the optimal order decision and the coordination of the buyback contract. The results show that the buyback contract can always coordinate the supply chain under the same condition; both the retailer's first order quantity and total order quantity decrease with wholesale price and increase with buyback price, but the retailer's fairness-concern behavior aggravates this trend.

KEYWORDS:

Disclosure statement

We declare that there is no conflict of interest regarding the publication of the paper ‘Buyback Contract Coordination in Supply Chain with Fairness Concern under Demand Updating’.

A2. Proof of Proposition 2

Similar to proof of Proposition 1, we can prove the Proposition 2 by backward induction. According to EquationEq. (6(6) $E (π_{l r 2}) = p E [min (q_{l}, x_{2})] - v E [max (x_{2} - q_{l}, 0)] + b E [max (q_{l} - x_{2}, 0)] - w q_{l 1} - (w + c_{Δ}) (q_{l} - q_{l 1}) = - (p + v - b) \int_{0}^{q_{l}} F_{2} (x_{2}) d x_{2} + (p + v - w) q_{l} - c_{Δ} (q_{l} - q_{l 1}) - v μ_{2}$ (6) ), the value of ${\hat{q}}_{l 1}$ and $μ_{2}$ is known, and the derivation of EquationEq. (6)(6) $E (π_{l r 2}) = p E [min (q_{l}, x_{2})] - v E [max (x_{2} - q_{l}, 0)] + b E [max (q_{l} - x_{2}, 0)] - w q_{l 1} - (w + c_{Δ}) (q_{l} - q_{l 1}) = - (p + v - b) \int_{0}^{q_{l}} F_{2} (x_{2}) d x_{2} + (p + v - w) q_{l} - c_{Δ} (q_{l} - q_{l 1}) - v μ_{2}$ (6) on the $q_{l}$ is:

(A5)

E (π_{l r 2}) = - (p + v - b) \int_{0}^{q_{l}} F_{2} (x_{2}) d x_{2} + (p + v - w) q_{l} - c_{Δ} (q_{l} - q_{l 1}) - v μ_{2}

(A5)

$\frac{d E (π_{l r 2})}{d q_{l}} = - (p + v - b) F_{2} (q_{l}) + p + v - w - c_{Δ}$ , $\frac{d^{2} E (π_{l r 2})}{d {q^{2}}_{l}} = - (p + v - b) f_{2} (q_{l}) < 0$

So the total optimal order quantity $q_{l}^{'}$ in the second stage exists and is unique. Let $\frac{d E (π_{l r 2})}{d q_{l}} = 0$ , we can get the total optimal order quantity as $q_{l}^{'} = μ_{2} + σ_{2} Φ^{- 1} (a)$ ( $a = \frac{p + v - w - c_{Δ}}{p + v - b}$ ). For the retailer’s order quantity is $q_{l 2} = max (0, {q^{'}}_{l} - q_{l 1})$ in the second stage, so there are two conditions as following:

(1) If $q_{l}^{'} - q_{l 1} \leq 0$ , then $q_{l 2} = 0$ , i.e. $μ_{2} \leq q_{l 1} - σ_{2} Φ^{- 1} (a)$ , insert $q_{l 2} = 0$ into Eq. (A5):

(A6)

E (π_{l r 2}) |_{q_{l 2} = 0} = - (p + v - b) \int_{0}^{q_{l 1}} F_{2} (x_{2}) d x_{2} + (p + v - w) q_{l 1} - v μ_{2}

(A6)

(2) If $q_{l}^{'} - q_{l 1} > 0$ , then $q_{l 2} = q_{l}^{'} - q_{l 1}$ i.e. $μ_{2} > q_{l 1} - σ_{2} Φ^{- 1} (a)$ , insert $q_{l 2} = q_{l}^{'} - q_{l 1}$ into Eq. (A5):

(A7)

E (π_{l r 2}) |_{q_{l 2} = {q^{'}}_{l} - q_{l 1}} = - (p + v - b) \int_{0}^{{q^{'}}_{l}} F_{2} (x_{2}) d x_{2} + (p + v - w) q_{l}^{'} - c_{Δ} ({q^{'}}_{l} - q_{l 1}) - v μ_{2}

(A7)

The expected profit in the first stage is as following:

(A8)

E (π_{l r 1}) = \int_{- \infty}^{q_{l 1} - σ_{2} Φ^{- 1} (a)} E (π_{l r 2}) |_{q_{l 2} = 0} g (μ_{2}) d μ_{2} + \int_{q_{l 1} - σ_{2} Φ^{- 1} (a)}^{+ \infty} E (π_{l r 2}) |_{q_{l 2} = {q^{'}}_{l} - q_{l}} g (μ_{2}) d μ_{2}

(A8)

For $\frac{d E (π_{l r 1})}{d q_{l 1}} = c_{Δ} + (p + v - w - c_{Δ}) A_{l} - (p + v - b) B_{l}$ , $\frac{d^{2} E (π_{l r 1})}{d q_{l 1}^{2}} = - \frac{p + v - b}{σ_{1}} C_{l} < 0$ .

$A_{l} = Φ (\frac{q_{l 1} - σ_{2} Φ^{- 1} (a) - μ_{1}}{σ})$ , $B_{l} = \int_{- \infty}^{\frac{q_{l 1} - σ_{2} Φ^{- 1} (a) - μ_{1}}{σ}} Φ (\frac{q_{l 1} - μ_{1} - σ γ}{σ_{2}}) φ (γ) d γ$ , $C_{l} = φ (\frac{q_{l 1} - μ_{1}}{σ_{1}}) Φ (\frac{q_{l 1} σ_{2}^{2} - σ_{2} σ_{1}^{2} Φ^{- 1} (a) - μ_{1} σ_{1}^{2}}{σ σ_{1} σ_{2}})$ .

$E (π_{l r 1})$ is strictly concave function of $q_{l 1}$ , so there is unique optimal $q_{l 1}^{*}$ maximising the expected profit $E (π_{l r 1})$ in the first stage. Let $\frac{d E (π_{l r 1})}{d q_{l 1}} = 0$ , we can get the optimal order quantity $q_{l 1}^{*}$ , and then we can get the real value of signal $x_{e z}$ , updating $μ_{2}$ as $μ_{2 z}$ , so the optimal order quantity of retailer is $q_{l}^{*} = max ({q^{'}}_{l}, q_{l 1}^{*})$ .

A3. Proof of Proposition 3

According to EquationEq. (15(15) $E (u_{n r 2}) = (1 + λ) [- (p + v - b) \int_{0}^{q_{n}} F_{2} (x_{2}) d x_{2} + (p + v - w) q_{n} - c_{Δ} (q_{n} - q_{n 1}) - v μ_{2 z}] - λ [(w - c) q_{n} - b \int_{0}^{q_{n}} F_{2} (x_{2}) d x_{2}]$ (15) ), the first and second derivation of $E (u_{n r 2})$ on $q_{n}$ is as following:

\frac{d E (u_{n r 2})}{d q_{n}} = - [(1 + λ) (p + v - b) - λ b] F_{2} (q_{n}) + (1 + λ) (p + v - w - c_{Δ}) - λ (w - c)

\frac{d^{2} E (u_{n r 2})}{d q_{n}^{2}} = - [(1 + λ) (p + v - b) - λ b] f_{2} (q_{n}) < 0

So the total optimal order quantity $q_{n}^{'}$ in the second stage exists and is unique to maximise his expected utility. Let $\frac{d E (u_{n r 2})}{d q_{n}} = 0$ , we can get the total optimal order quantity of retailer is $q_{n}^{'} = μ_{2 z} + σ_{2} Φ^{- 1} (z (λ))$ ( $z (λ) = \frac{(1 + λ) (p + v - w - c_{Δ}) - λ (w - c)}{(1 + λ) (p + v - b) - λ b}$ ). The market signal is unknown in the first stage, so $μ_{2}$ is a random variable and $q_{n 2} = max (0, {q^{'}}_{n} - q_{n 1})$ . The proof of Proposition 2 can be discussed in two conditions as following:

(1) If $q_{n}^{'} - q_{n 1} \leq 0$ , then $q_{n 2} = 0$ , i.e. $μ_{2} \leq q_{n 1} - σ_{2} Φ^{- 1} (z (λ))$ , insert $q_{n 2} = 0$ into EquationEq. (15)(15) $E (u_{n r 2}) = (1 + λ) [- (p + v - b) \int_{0}^{q_{n}} F_{2} (x_{2}) d x_{2} + (p + v - w) q_{n} - c_{Δ} (q_{n} - q_{n 1}) - v μ_{2 z}] - λ [(w - c) q_{n} - b \int_{0}^{q_{n}} F_{2} (x_{2}) d x_{2}]$ (15) :

(A9)

E (u_{n r 2}) |_{q_{n 2} = 0} = (1 + λ) [- (p + v - b) \int_{0}^{q_{n 1}} F_{2} (x_{2}) d x_{2} + (p + v - w) q_{n 1} - v μ_{2 z}] - λ [(w - c) q_{n 1} - b \int_{0}^{q_{n 1}} F_{2} (x_{2}) d x_{2}]

(A9)

(2) If $q_{n 2} = q_{n}^{'} - q_{n 1}$ , then $q_{n}^{'} - q_{n 1} > 0$ , i.e. $μ_{2} > q_{n 1} - σ_{2} Φ^{- 1} (z (λ))$ , insert $q_{n 2} = q_{n}^{'} - q_{n 1}$ into EquationEq. (15)(15) $E (u_{n r 2}) = (1 + λ) [- (p + v - b) \int_{0}^{q_{n}} F_{2} (x_{2}) d x_{2} + (p + v - w) q_{n} - c_{Δ} (q_{n} - q_{n 1}) - v μ_{2 z}] - λ [(w - c) q_{n} - b \int_{0}^{q_{n}} F_{2} (x_{2}) d x_{2}]$ (15) :

(A10)

E (u_{n r 2}) |_{q_{n 2} = {q^{'}}_{n} - q_{n 1}} = (1 + λ) [- (p + v - b) \int_{0}^{{q^{'}}_{n}} F_{2} (x_{2}) d x_{2} + (p + v - w) q_{n}^{'} - v μ_{2 z} - c_{Δ} ({q^{'}}_{n} - q_{n 1})] - λ [(w - c) {q^{'}}_{n} - b \int_{0}^{{q^{'}}_{n}} F_{2} (x_{2}) d x_{2}]

(A10)

EquationEquations (A9)(A9) $E (u_{n r 2}) |_{q_{n 2} = 0} = (1 + λ) [- (p + v - b) \int_{0}^{q_{n 1}} F_{2} (x_{2}) d x_{2} + (p + v - w) q_{n 1} - v μ_{2 z}] - λ [(w - c) q_{n 1} - b \int_{0}^{q_{n 1}} F_{2} (x_{2}) d x_{2}]$ (A9) and (A10) denote the expected utility of retailer under different conditions, and then the expected utility is:

(A11)

E (u_{n r 1}) = \int_{- \infty}^{q_{n 1} - σ_{2} Φ^{- 1} (z (λ))} E (u_{n r 2}) |_{q_{n 2} = 0} g (μ_{2}) d μ_{2} + \int_{q_{n 1} - σ_{2} Φ^{- 1} (z (λ))}^{+ \infty} E (u_{n r 2}) |_{q_{n 2} = {q^{'}}_{n} - q_{n 1}} g (μ_{2}) d μ_{2}

(A11)

The derivation of Eq. (A11) on $q_{n 1}$ is:

$\frac{d E (u_{n r 1})}{d q_{n 1}} = (1 + λ) c_{Δ} + [(1 + λ) (p + v - w - c_{Δ}) - λ (w - c)] A_{n} - [(1 + λ) (p + v - b) - λ b] B_{n}$ , $\frac{d^{2} E (u_{n r 1})}{d q_{n 1}^{2}} = - \frac{[(1 + λ) (p + v - b) - λ b]}{σ_{1}} C_{n}$

$A_{n} = Φ (\frac{q_{n 1} - σ_{2} Φ^{- 1} (z (λ)) - μ_{1}}{σ}), B_{n} = \int_{- \infty}^{\frac{q_{n 1} - σ_{2} Φ^{- 1} (z (λ)) - μ_{1}}{σ}} Φ (\frac{q_{n 1} - μ_{1} - σ γ}{σ_{2}}) φ (γ) d γ$ , $C_{n} = φ (\frac{q_{n 1} - μ_{1}}{σ_{1}}) Φ (\frac{q_{n 1} σ_{2}^{2} - σ_{2} σ_{1}^{2} Φ^{- 1} (z (λ)) - μ_{1} σ_{1}^{2}}{σ σ_{1} σ_{2}})$ .

$E (u_{n r 1})$ is strictly concave function of $q_{n 1}$ , so there is unique optimal $q_{n 1}^{*}$ maximising the expected profit $E (u_{n r 1})$ in the first stage. Let $\frac{d E (u_{n r 1})}{d q_{n 1}} = 0$ , we can compute $q_{n 1}^{*}$ and then we can get the real value of signal $x_{e z}$ , and update $μ_{2}$ as $μ_{2 z}$ , so the optimal order quantity of retailer is $q_{n}^{*} = max ({q^{'}}_{n}, q_{n 1}^{*})$ .

A4. Proof of Nature 1

The order quantity of retailer in the first stage can be denoted as following:

(A12)

(1 + λ) c_{Δ} + [(1 + λ) (p + v - w - c_{Δ}) - λ (w - c)] A_{n} - [(1 + λ) (p + v - b) - λ b] B_{n} = 0

(A12)

We can simplify Eq. (A12) as following:

(A13)

\frac{d q_{n 1}^{*}}{d λ} = \frac{[c_{Δ} + (p + v - c - c_{Δ}) A_{n}] b - (p + v) (w - c) A_{n}}{{[(1 + λ) (p + v - b) - λ b]}^{2} C_{n}} σ_{1}

(A13)

According to Eq. (A13), if $b < \frac{(p + v) (w - c) A_{n}}{c_{Δ} + (p + v - c - c_{Δ}) A_{n}}$ , then $\frac{d q_{n 1}^{*}}{d λ} < 0$ ; If $b > \frac{(p + v) (w - c) A_{n}}{c_{Δ} + (p + v - c - c_{Δ}) A_{n}}$ , then $\frac{d q_{n 1}^{*}}{d λ} > 0$ . For $q_{n}^{*} = μ_{2 z} + σ_{2} Φ^{- 1} (z (λ))$ , we can get $\frac{d q_{n}^{*}}{d λ} = \frac{- (p + v) (w - c) + (p + v - c - c_{Δ}) b}{{[(1 + λ) (p + v - b) - λ b]}^{2} f_{2} (q_{n}^{*})}$ , and if $b < \frac{(p + v) (w - c)}{p + v - c - c_{Δ}}$ , then $\frac{d q_{n}^{*}}{d λ} < 0$ ; if $b > \frac{(p + v) (w - c)}{p + v - c - c_{Δ}}$ , then $\frac{d q_{n}^{*}}{d λ} > 0$ . It is easy to prove $\frac{(p + v) (w - c) A_{n}}{c_{Δ} + (p + v - c - c_{Δ}) A_{n}} < \frac{(p + v) (w - c)}{p + v - c - c_{Δ}}$ . If $b < \frac{(p + v) (w - c) A_{n}}{c_{Δ} + (p + v - c - c_{Δ}) A_{n}}$ , then $\frac{d q_{n 1}^{*}}{d λ} < 0$ and $\frac{d q_{n}^{*}}{d λ} < 0$ ; If $\frac{(p + v) (w - c) A_{n}}{c_{Δ} + (p + v - c - c_{Δ}) A_{n}} < b < \frac{(p + v) (w - c)}{p + v - c - c_{Δ}}$ , then $\frac{d q_{n 1}^{*}}{d λ} > 0$ and $\frac{d q_{n}^{*}}{d λ} < 0$ ; If $b > \frac{(p + v) (w - c)}{p + v - c - c_{Δ}}$ , then $\frac{d q_{n 1}^{*}}{d λ} > 0$ and $\frac{d q_{n}^{*}}{d λ} > 0$ .

A5. Proof of Conclusion 1

(1) The proof in the case when the retailer is complete rationality.

In order to coordinate the supply chain, it is necessary to keep the order quantity and total order quantity of the retailer consistent with the order quantity under the centralised decision-making model in the first stage when the real value of market demand signal $x_{e z}$ is updated, i.e. $q_{l 1}^{*} = {\hat{q}}_{l 1}^{*}$ and $q_{l}^{'} = {\hat{q}}_{l}^{'}$ . For $q_{l}^{'} = {\hat{q}}_{l}^{'}$ and $w = \frac{p + v - c - c_{Δ}}{p + v} b + c$ , we can get the following equation:

(A14)

c_{Δ} + (p + v - c - c_{Δ}) {\hat{A}}_{l} - (p + v) {\hat{B}}_{l} = 0

(A14)

For $q_{l 1}^{*} = {\hat{q}}_{l 1}^{*}$ , then we can solve the Eq. (A14), and get $w = \frac{(p + v - c - c_{Δ}) b}{p + v} + c + \frac{b c_{Δ}}{(p + v) A_{l}}$

In order to coordinate the supply chain, it is necessary to let $q_{l 1}^{*} = {\hat{q}}_{l 1}^{*}$ and $q_{l}^{'} = {\hat{q}}_{l}^{'}$ , i.e. $w = \frac{p + v - c - c_{Δ}}{p + v} b + c$ and $w = \frac{(p + v - c - c_{Δ}) b}{p + v} + c + \frac{b c_{Δ}}{(p + v) A_{l}}$ , so $c_{Δ} = 0$ and $w = \frac{p + v - c}{p + v} b + c$ , i.e. buyback contract can coordinate the supply chain.

(2) The proof in the case when the retailer cares about fairness.

Similar to the proof in the case when the retailer is complete rationality, it is necessary to let $q_{n 1}^{*} = {\hat{q}}_{n 1}^{*}$ and $q_{n}^{'} = {\hat{q}}_{n}^{'}$ , i.e. we can prove that $w = \frac{(p + v - c - c_{Δ}) b}{p + v} + c + \frac{b c_{Δ}}{(p + v) A_{n}}$ , and when $c_{Δ} = 0$ , $w = \frac{p + v - c}{p + v} b + c$ . So when the retailer cares about fairness, the buyback contract can coordinate the supply chain in the same condition.

A6. Proof of Conclusion 2

(1) The proof in the case when the retailer is complete rational.

According to Proposition 2, ① $q_{l}^{*} = q_{l}^{'}$ . when the optimal order quantity is $q_{l}^{*} = μ_{2 z} + σ_{2} Φ^{- 1} (a)$ and the retailer is complete rationality, and $\frac{\partial q_{l}^{*}}{\partial w} = - \frac{1}{(p + v - b) f_{2} (q_{l}^{*})} < 0$ , $\frac{\partial q_{l}^{*}}{\partial b} = \frac{p + v - w - c_{Δ}}{(p + v - b) f_{2} (q_{l}^{*})} > 0$ . ② $q_{l}^{*} = q_{l 1}^{*}$ . when the optimal order quantity subject to $c_{Δ} + (p + v - w - c_{Δ}) A_{l} - (p + v - b) B_{l} = 0$ , it is easy to obtain $\frac{\partial q_{l}^{*}}{\partial w} = - \frac{σ_{1} A_{l}}{(p + v - b) C_{l}}$ , $\frac{\partial q_{l}^{*}}{\partial b} = \frac{σ_{1} B_{l}}{(p + v - b) C_{l}}$ , and for $p + v - b > 0$ , $A_{l} > 0$ , $C_{l} > 0$ and $B_{l} > 0$ , therefore $\frac{\partial q_{l}^{*}}{\partial w} < 0$ , $\frac{\partial q_{l}^{*}}{\partial b} > 0$ .

(2) The proof in the case when the retailer cares about fairness.

According to Proposition 3, ① $q_{n}^{*} = q_{n}^{'}$ . when the optimal order quantity is $q_{n}^{*} = μ_{2 z} + σ_{2} Φ^{- 1} (z (λ))$ , and the retailer is fairness concern, and $\frac{\partial q_{n}^{*}}{\partial w} = - \frac{1 + 2 λ}{[(1 + λ) (p + v - b) - λ b] f_{2} (q_{n}^{*})} < 0$ , $\frac{\partial q_{n}^{*}}{\partial b} = \frac{[(1 + λ) (p + v - w - c_{Δ}) - λ (w - c)] (1 + 2 λ)}{[(1 + λ) (p + v - b) - λ b] f_{2} (q_{n}^{*})} > 0$ . ② $q_{n}^{*} = q_{n 1}^{*}$ . The optimal order quantity is subjected to Eq. (A15)

(A15)

(1 + λ) c_{Δ} + [(1 + λ) (p + v - w - c_{Δ}) - λ (w - c)] A_{n} - [(1 + λ) (p + v - b) - λ b] B_{n} = 0

(A15)

$\frac{\partial q_{n}^{*}}{\partial w} = - \frac{(1 + 2 λ) σ_{1} A_{n}}{[(1 + λ) (p + v - b) - λ b] C_{n}}$ , $\frac{\partial q_{n}^{*}}{\partial b} = \frac{(1 + 2 λ) σ_{1} B_{n}}{[(1 + λ) (p + v - b) - λ b] C_{n}}$ (A16)

For $(1 + λ) (p + v - b) - λ b > 0$ , $A_{n} > 0$ , $C_{n} > 0$ and $B_{n} > 0$ , then $\frac{\partial q_{n}^{*}}{\partial w} < 0$ , $\frac{\partial q_{n}^{*}}{\partial b} > 0$ .

Furthermore, it is easy to prove $|\frac{\partial q_{n}^{*}}{\partial w}| > |\frac{\partial q_{l}^{*}}{\partial w}|$ , $|\frac{\partial q_{n}^{*}}{\partial b}| > |\frac{\partial q_{l}^{*}}{\partial b}|$ . So both the optimal order quantity in the first stage and total order quantity are negatively correlated with wholesale price, positively correlated with buyback price, and the fairness concern aggravate these changing trend.

A7. Proof of Conclusion 3

(1) The proof in the case when the retailer is complete rational.

According to Proposition 2, the optimal order quantity of the retailer is subjected to $c_{Δ} + (p + v - w - c_{Δ}) A_{l} - (p + v - b) B_{l} = 0$ , and then $\frac{d q_{l 1}^{*}}{d c_{Δ}} = \frac{(1 - A_{l}) σ_{1}}{(p + v - b) C_{l}}$ ( $1 - A_{l} > 0$ , $p + v - b > 0$ , $C_{l} > 0$ ), i.e. $\frac{d q_{l 1}^{*}}{d c_{Δ}} > 0$ . Besides, the total order quantity of retailer is $q_{l}^{*} = μ_{2 z} + σ_{2} Φ^{- 1} (a)$ , so $\frac{d q_{l}^{*}}{d c_{Δ}} = - \frac{1}{(p + v - b) f_{2} (q_{l}^{*})} < 0$

(2) The proof in the case when the retailer cares about fairness.

According to Proposition 3, the optimal order quantity of the retailer in the first stage subject to Eq. (A17)

(A17)

(1 + λ) c_{Δ} + [(1 + λ) (p + v - w - c_{Δ}) - λ (w - c)] A_{n} - [(1 + λ) (p + v - b) - λ b] B_{n} = 0

(A17)

It is easy to compute $\frac{d q_{n 1}^{*}}{d c_{Δ}} = \frac{(1 + λ) (1 - A_{n}) σ_{1}}{[(1 + λ) (p + v - b) - λ b] C_{n}}$ ( $1 - A_{n} > 0$ , $(1 + λ) (p + v - b) - λ b > 0$ , $C_{n} > 0$ ), and then $\frac{d q_{n 1}^{*}}{d c_{Δ}} > 0$ . Besides, the total order quantity of retailer is $q_{n}^{*} = μ_{2 z} + σ_{2} Φ^{- 1} (z (λ))$ , so $\frac{d q_{n}^{*}}{d c_{Δ}} = - \frac{1 + λ}{[(1 + λ) (p + v - b) - λ b] f_{2} (q_{n}^{*})} < 0$ .

Additional information

Funding

This work was supported by Chinese National Natural Science Foundation [No. 71572028], [No. 71872027] and [No. 71572030], and the funder is Yunfei Shao.

Buyback contract coordination in supply chain with fairness concern under demand updating

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Buyback contract coordination in supply chain with fairness concern under demand updating

ABSTRACT

Disclosure statement

A2. Proof of Proposition 2

A3. Proof of Proposition 3

A4. Proof of Nature 1

A5. Proof of Conclusion 1

A6. Proof of Conclusion 2

A7. Proof of Conclusion 3

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Funding

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