Value of loan credit insurance in the capital-constrained supply chain

Abstract

Supply chain finance (SCF) solutions open financing channels for small and medium-sized suppliers. However, in uncertain market conditions, payment defaults by supply chain members may affect SCF systems sustainability. To mitigate default risks, financial service providers utilize loan credit insurance (LCI) to transfer the risk to insurers. LCI, purchased by lenders, covers unpaid losses resulting from debtor insolvency, bankruptcy, or political upheaval. In case retailers fail to meet payment obligations, insurers will compensate the lender within the policy terms. Using a game-theoretical approach, we examine LCI’s value for lenders and capital-constrained suppliers. We find that lenders benefit from LCI with higher insurance deductibles and lower insurer loading factors. Moreover, capital-constrained suppliers with higher capital investments and lower production costs can benefit from LCI. Furthermore, suppliers are more likely to obtain extra profits when the insurance scale and loading factor are relatively low. However, when the insurance scale and loading factor are relatively high, the supplier’s capacity investment efficiency is greater. Numerical analysis indicates that LCI with relatively high coverage is advantageous for lenders, particularly full coverage. Another numerical finding is that when the insurance scale is large and the loading factor is high, suppliers achieve greater cost contribution efficiency. Our study highlights LCI’s value in capital-constrained supply chains, benefiting both lenders and suppliers by enhancing profits and encouraging suppliers to increase capital investment and reduce costs. It underscores the role of lender-purchased credit insurance in SCF risk management, providing theoretical support and practical guidance for LCI implementations in commercial financial institutions.

Appendices: Proofs

1.1 Proof of Lemma 1

According to Eq. (2) and taking the first-order derivative of $\Pi _s$ with respect to q, we obtain

$$\begin{aligned} \frac{d\Pi _s}{dq}=w\bar{F}\left( q\right) -\frac{c}{1-r}\bar{F}\left( {\hat{x}}_3\right) . \end{aligned}$$

Subsequently, according to the supplier’s first-order condition (FOC), let $d\Pi _s/dq=0$; we then have $w (1-r) \bar{F}\left( q\right) =c \bar{F}\left( {\hat{x}}_3\right) .$ We can prove that $\bar{F}\left( q\right) < \bar{F}\left( {\hat{x}}_3\right)$ because $F\left( \cdot \right)$ is a strictly increasing function and $q > B/w$. Thus, $c< w (1-r) < w$.

Furthermore, by taking the second-order derivative of $\Pi _s$ with respect to q, we have

$$\begin{aligned} \frac{d^2\Pi _s}{dq^2}={\left( \frac{c}{1-r}\right) }^2f\left( \hat{x}_3\right) -wf\left( q\right) . \end{aligned}$$

By substituting the supplier’s FOC into ${d^2\Pi _s}/{dq^2}$, we have

$$\begin{aligned} \begin{aligned} \frac{d^2\Pi _s}{dq^2}&=\bar{F}\left( \hat{x}_3\right) \left[ \left( \frac{c}{1-r}\right) ^2 \frac{h\left( \hat{x}_3\right) }{w}-w h\left( q\right) {\frac{\bar{F}\left( q\right) }{\bar{F}\left( \hat{x}_3\right) }}\right] \\&=\bar{F}\left( \hat{x}_3\right) \frac{c}{1-r} \left[ \frac{c}{w\left( 1-r\right) } h\left( \hat{x}_3\right) -h\left( q\right) \right] . \end{aligned} \end{aligned}$$

Based on the IFR assumption of the demand function, we have $h\left( \hat{x}_3\right) < h\left( q\right)$ because $\hat{x}_3 < q$. We have derived that $w (1-r) > c$, thus, we have ${d^2\Pi _s}/{dq^2}<0$—$\Pi _s$ is a concave function in terms of q at $d\Pi _s/dq=0$. Therefore, the supplier’s best response under the given w and r is

$$\begin{aligned} w\bar{F} \left( q^*\right) =\frac{c}{1-r}\bar{F} \left( {\hat{x}}_3\right) \end{aligned}$$

(A.1)

1.2 Proof of Lemma 2

According to Eq. (3), and the end of Eq. (A.1), by taking the first-order derivative of $\Pi _r$ with respect to w yields

$$\begin{aligned} \frac{d\Pi _r}{dw}=\left( 1-w\right) \frac{\partial q^*}{\partial w} \bar{F}\left( q^*\right) - \int _{0}^{q^*}\bar{F}\left( x\right) dx. \end{aligned}$$

we denote $\Omega _1=\frac{\partial q^*}{\partial w}$. By differentiating both sides of Eq.(A.1) with respect to w, we have

$$\begin{aligned} \bar{F}\left( q^*\right) - w \Omega _1 f\left( q^*\right) = -\frac{c}{1-r} \frac{\partial \hat{x}_3}{\partial w} f\left( \hat{x}_3\right) , \end{aligned}$$

where $\frac{\partial \hat{x}_3}{\partial w}=\left( \frac{c}{1-r}\Omega _1-\frac{B}{w}\right) /w$, Substituting the supplier’s FOC for simplicity, we have

$$\begin{aligned} \begin{aligned} {\Omega }_1&=\frac{\bar{F}\left( q^*\right) -\frac{cB}{(1-r)w^2} f\left( \hat{x}_3\right) }{wf\left( q^*\right) -\left( \frac{c}{1-r}\right) ^2\frac{f\left( \hat{x}_3\right) }{w}} =\frac{1-\frac{cB}{(1-r)w^2}\frac{\bar{F}\left( \hat{x}_3\right) }{\bar{F}(q^*)}h\left( \hat{x}_3\right) }{w h(q^*)-\left( \frac{c}{1-r}\right) ^2\frac{h\left( \hat{x}_3\right) }{w} \frac{\bar{F}\left( \hat{x}_3\right) }{\bar{F}(q^*)}}\\&=\frac{1-H\left( {\hat{x}}_3\right) }{w h\left( q^*\right) -c h\left( {\hat{x}}_3\right) /(1-r)}. \end{aligned} \end{aligned}$$

Based on the properties of the IFR distribution, we have $1-H\left( {\hat{x}}_3\right) > 0$. As we have proven that $w (1-r) > c$ and $h\left( \hat{x}_3\right) < h\left( q\right)$ in the proof of Lemma 1, we obtain $w h\left( q^*\right) -c h\left( {\hat{x}}_3\right) /(1-r)>0$. Therefore, $\Omega _1=\frac{\partial q^*}{\partial w}>0$.

Subsequently, according to FOC and letting $d\Pi _r/dw=0$, the retailer’s optimal procurement price satisfies

$$\begin{aligned} \left( 1-w^*\right) \frac{\partial q^*}{\partial w} \bar{F}\left( q^*\right) = \int _{0}^{q^*}\bar{F}\left( x\right) dx. \end{aligned}$$

(A.2)

1.3 Proof of Lemma 3

Similar to the proof of Lemma 2, according to Eq. (4), and Eq. (A.1), taking the first-order derivative of $\Pi _f$ with respect to r yields

$$\begin{aligned} \frac{d\Pi _f}{dr} = \frac{\partial B}{\partial r} \{r + \theta \left[ F\left( \hat{x}_1\right) -F\left( \hat{x}_2\right) \right] - F\left( \hat{x}_3\right) \} + B, \end{aligned}$$

where $\frac{\partial B}{\partial r}= \frac{c}{1-r} \frac{\partial q^*}{\partial r} + \frac{B}{1-r}$.

For notational simplification, we denote $\Omega _2=\frac{\partial B}{\partial r}$ and $\Omega _3=\frac{\partial q^*}{\partial r}$. By differentiating both sides of Eq.(A.1) with respect to r, we have

$$\begin{aligned} -w \Omega _3 f\left( q^*\right) = \frac{c}{(1-r)^2} \bar{F}\left( \hat{x}_3\right) - \frac{c}{w(1-r)} \Omega _2 f\left( \hat{x}_3\right) . \end{aligned}$$

Substituting the supplier’s FOC for simplicity, we have

$$\begin{aligned} \begin{aligned} \Omega _3&=\frac{c \bar{F}\left( \hat{x}_3\right) - cB f\left( \hat{x}_3\right) /w}{c^2 f\left( \hat{x}_3\right) /w- w\left( 1-r\right) ^2 f(q^*)} =\frac{c-cB h\left( {\hat{x}}_3\right) /w}{c^2 h\left( {\hat{x}}_3\right) /w - w\left( 1-r\right) ^2 h(q^*) \frac{\bar{F}(q^*)}{\bar{F}\left( \hat{x}_3\right) }}\\&=\frac{1-H\left( {\hat{x}}_3\right) }{{ch\left( {\hat{x}}_3\right) }/{w}-(1-r)h\left( q^*\right) }. \end{aligned} \end{aligned}$$

Subsequently, according to the supplier’s FOC condition and letting ${d\Pi _f}/{dr}=0$, the lender’s optimal interest rate satisfies

$$\begin{aligned} r^*=\{F\left( {\hat{x}}_3\right) +\theta \left[ F\left( {\hat{x}}_2\right) -F\left( {\hat{x}}_1\right) \right] \}-{{B}/{\Omega _2}} \end{aligned}$$

(A.3)

1.4 Proof of Corollary 1

• (1) Based on $\Omega _1$ and $\Omega _3$ that we deduced in the proof of Lemma 2 and Lemma 3, we find the denominators of $\Omega _1$ and $\Omega _3$ have the opposite sign because $\Omega _1=-(1-r)\Omega _3/w$. In the proof of Lemma 2, we have $\Omega _1>0$. Thus, we can easily verify $\Omega _3=\frac{\partial q^*}{\partial r}<0$.

• (2) Substituting the formula of $\Omega _3$ into $\Omega _2=\frac{\partial B}{\partial r}= \frac{c}{1-r} \frac{\partial q^*}{\partial r} + \frac{B}{1-r}$, we have

  $$\begin{aligned} \begin{aligned} \Omega _2&=\frac{c\left[ 1-H\left( \hat{x}_3\right) \right] }{c(1-r)h\left( \hat{x}_3\right) /w-(1-r)^2h(q^*)}+\frac{B}{1-r}\\&=\frac{c-(1-r)Bh(q^*)}{(1-r)\left[ ch\left( \hat{x}_3\right) /w-(1-r)h(q^*)\right] } \\&=\frac{c\left[ 1-(q^*-k/c)h(q^*)\right] }{(1-r)\left[ ch\left( \hat{x}_3\right) /w-(1-r)h(q^*)\right] }\\&>\frac{c\left[ 1-H(q^*)\right] }{(1-r)\left[ ch\left( \hat{x}_3\right) /w-(1-r)h(q^*)\right] }. \end{aligned} \end{aligned}$$

  Based on the properties of the IFR distribution, we have $1-H(q^*) > 0$. Since we have proved that $w (1-r) > c$ and $h\left( \hat{x}_3\right) < h\left( q\right)$ in the proof of Lemma 1, we obtain $ch\left( \hat{x}_3\right) /w-(1-r)h(q^*)<0$. Thus, we have $\Omega _2=\frac{\partial B}{\partial r}<0$. Next, we simplify Eq.(A.3) into $B=\Omega _2 \{F\left( \hat{x}_3\right) -\theta \left[ F\left( \hat{x}_1\right) -F\left( \hat{x}_2\right) \right] -r^*\}$. We can deduce that $F\left( \hat{x}_3\right) -\theta \left[ F\left( \hat{x}_1\right) -F\left( \hat{x}_2\right) \right] -r^*<0$ because $\Omega _2<0$ and $B>0$ hold. Since $\hat{x}_1<\hat{x}_2$ and the increasing demand CDF, we have $F\left( \hat{x}_1\right) -F\left( \hat{x}_2\right) <0$. Thus, we obtain the restriction for the loading factor, $\theta < \frac{r^*-F\left( \hat{x}_3\right) }{F\left( \hat{x}_2\right) -F\left( \hat{x}_1\right) }$.

• (3) Taking the partial derivative with respect to k on both sides of Eq.(A.1), we have

  $$\begin{aligned} -w^2\frac{\partial q^*}{\partial k} f(q^*)=-\left( \frac{c}{1-r}\right) ^2 \frac{\partial q^*}{\partial k} f\left( \hat{x}_3\right) + \frac{c}{(1-r)^2} f\left( \hat{x}_3\right) \end{aligned}$$

  . Additionally, substituting Eq.(A.1) into the above formula and simplifying, we have

  $$\begin{aligned} \begin{aligned} \frac{\partial q^*}{\partial k}&=\frac{cf\left( \hat{x}_3\right) }{c^2 f\left( \hat{x}_3\right) - w^2 (1-r)^2 f(q^*)} =\frac{cf\left( \hat{x}_3\right) }{\bar{F}\left( \hat{x}_3\right) \left[ c^2 h\left( \hat{x}_3\right) -w^2 (1-r)^2 h(q^*) \frac{\bar{F}(q^*)}{\bar{F}\left( \hat{x}_3\right) }\right] }\\&=\frac{h\left( \hat{x}_3\right) }{ch\left( \hat{x}_3\right) -w(1-r)h(q^*)} \end{aligned} \end{aligned}$$

  Therefore, we have ${\partial q^*}/{\partial k} <0$ for $h\left( \hat{x}_3\right) >0$ and $ch\left( \hat{x}_3\right) -w(1-r)h(q^*)<0$.

1.5 Proof of Proposition 1

According to Eq. (4) and the lender’s profit function in the benchmark, we calculate the lender’s profit difference between the LCI scheme and benchmark.

$$\begin{aligned} \Delta \Pi _f=\Pi _f^*- \Pi _f^{N*}=Br^*- w^*\theta \int ^{\hat{x} _2}_{\hat{x}_1} {F\left( x\right) }dxw^*\ int^{\hat{x}_3} _0{F\left( x\right) }dx - B^N {r^N}^* + {w^N} ^*\int ^{{\hat{x}}^N_3} _0 F\left( x\right) dx. \end{aligned}$$

For notational simplification, let $Y(x)=\int ^x_0 F(x)dx$, which is a monotonically increasing function because $dY(x)/dx=F(x)>0$. Thus, the necessary condition for the lender to benefit from LCI (i.e., $\Delta \Pi _f \ge 0$) is

$$\begin{aligned} Br^*-w^*\theta \left[ Y\left( \hat{x} _2\right) -Y\left( \hat{x} _1\right) \right] - w^* Y\left( \hat{x}_3\right) >B^N {r^N}^* - {w^N} ^*Y\left( \hat{x} _3^N\right) . \end{aligned}$$

(A.4)

The condition Eq. (A.4) can be rewritten as $Y\left( \hat{x}_2\right) <\frac{1}{\theta w^*}\mathscr {A}_1+Y\left( \hat{x}_1\right)$. If $\frac{1}{\theta w^*}\mathscr {A}_1+Y\left( \hat{x}_1\right) \ge 0$, we have $\hat{x}_2<Y^{-1}\left[ \frac{1}{\theta w^*}\mathscr {A}_1+Y\left( \hat{x}_1\right) \right]$ since $Y^{-1}(x)$ has the same monotonicity as Y(x). Thus, the condition can be simplified to $d>B-w^*Y^{-1}\left[ \frac{1}{\theta w^*}\mathscr {A}_1+Y\left( \hat{x}_1\right) \right]$. If $\frac{1}{\theta w^*}\mathscr {A}_1+Y\left( \hat{x}_1\right) <0$, this condition does not hold because $Y\left( \hat{x}_2\right) \ge 0 > \frac{1}{\theta w^*}\mathscr {A}_1+Y\left( \hat{x}_1\right)$.

Hence, let $\mathscr {A}_1={\left[ Br^*-w^*Y\left( {\hat{x}}_3\right) -{B^N}{r^N}^*+w^*Y\left( {\hat{x}}^N_3\right) \right] }/{w^*\theta }+Y\left( {\hat{x}}_1\right)$, if $\mathscr {A}_1 \ge 0$. There exists a threshold

$$\begin{aligned} \hat{d}=B-w^*Y^{-1}\left( \mathscr {A}_1\right) , \end{aligned}$$

for which $d\ge \hat{d}$ is the value of LCI for the lender $\Delta \Pi _f \ge 0$. Otherwise, the value of LCI for the lender is $\Delta \Pi _f < 0$.

From the perspective of the loading factor, Eq. (A.4) can be rewritten as $w^*\theta \left[ Y\left( \hat{x} _2\right) -Y\left( \hat{x} _1\right) \right] \le Br^*- w^*Y\left( {\hat{x}} _3\right) -{B^N}{r^N} ^*+w{N^*} Y\left( {\hat{x}} ^N_3\right)$. Clearly, $\left[ Y\left( \hat{x}_2\right) -Y\left( \hat{x}_1\right) \right] >0$ because $\hat{x}_2>\hat{x}_1$. Then, the value-added condition can be simplified as $\theta \le \frac{Br^*- w^*Y\left( {\hat{x}} _3\right) -{B^N}{r^N} ^*+w^{N*} Y\left( {\hat{x}} ^N_3\right) }{w^* \left[ Y\left( \hat{x} _2\right) -Y\left( \hat{x} _1\right) \right] }$. Therefore, there exists a threshold

$$\begin{aligned} \hat{\theta }=\frac{Br^*- w^*Y\left( {\hat{x}} _3\right) -{B^N}{r^N} ^*+w^{N*} Y\left( {\hat{x}} ^N_3\right) }{w^* \left[ Y\left( \hat{x} _2\right) -Y\left( \hat{x} _1\right) \right] }, \end{aligned}$$

When $\theta \le \hat{\theta }$, the value of LCI for the lender is $\Delta \Pi _f \ge 0$. Otherwise, when $\theta > \hat{\theta }$, the value of LCI for the lender is $\Delta \Pi _f < 0$.

1.6 Proof of Proposition 2

According to Eq. (2) and the supplier’s profit function in the benchmark, and calculating the supplier’s profit difference between the LCI scheme and the benchmark, we obtain

$$\begin{aligned} \Delta \Pi _s=\Pi _s^*- \Pi _s^{N*} =w^*\int ^q_{\hat{x}_3} {\bar{F}\left( x\right) }dx- {w^N}^*\int ^{{q^N}^*} _{{\hat{x}}^N_3}\bar{F} \left( x\right) dx. \end{aligned}$$

To simplify the notation, let $S(x)=\int ^x_0 \bar{F}(x)dx$ be a monotonically increasing function because $dS(x)/dx=\bar{F}(x)>0$. Thus, the necessary condition for the supplier to benefit from LCI (i.e., $\Delta \Pi _s \ge 0$) is

$$\begin{aligned} w^* S\left( \hat{x}_3\right) < {w^N}^* S\left( \hat{x}_3^N\right) +w^* S\left( q^*\right) - {w^N}^* S\left( {q^N}^*\right) . \end{aligned}$$

(A.5)

If ${w^N}^* S\left( \hat{x}_3^N\right) +w^* S\left( q^*\right) - {w^N}^* S\left( {q^N}^*\right) \ge 0$, Eq. (A.5) can be rewritten as $B \le w^* S^{-1}\{S(q^*)-\frac{{w^N}^*}{w^*} \left[ S\left( {q^N}^*\right) -S\left( \hat{x}_3^N\right) \right] \}$ since $S^{-1}(x)$ has the same monotonicity as S(x). Thus, $k \ge cq^*-(1-r^*)w^* S^{-1}\{S(q^*)-\frac{{w^N}^*}{w^*} \left[ S\left( {q^N}^*\right) -S\left( \hat{x}_3^N\right) \right] \}$. If ${w^N}^* S\left( \hat{x}_3^N\right) +w^* S\left( q^*\right) - {w^N}^* S\left( {q^N}^*\right) < 0$; the condition in Eq.(A.5) does not hold, because $S\left( \hat{x}_3\right) \ge 0 > {w^N}^* S\left( \hat{x}_3^N\right) +w^* S\left( q^*\right) - {w^N}^* S\left( {q^N}^*\right)$.

Hence, let $\mathscr {A}_2=S(q^*)-\frac{{w^N}^*}{w^*} \left[ S\left( {q^N}^*\right) -S\left( \hat{x}_3^N\right) \right]$. If $\mathscr {A}_2 \ge 0$, there exists a threshold

$$\begin{aligned} \hat{k}=cq^*-(1-r^*)w^*S^{-1}\left( \mathscr {A}_2\right) , \end{aligned}$$

that satisfies $cq^*-\hat{k}>0$. When $k\ge \hat{k}$, the value of LCI for the supplier $\Delta \Pi _s \ge 0$. Otherwise, the value of LCI for the supplier $\Delta \Pi _s < 0$.

1.7 Proof of Corollary 2

• (1) From the lender’s FOC condition in Eq. (A.3), we have

  $$\begin{aligned} 1-r^*=B/\Omega _2+\bar{F}\left( \hat{x}_3\right) -\theta \left[ \bar{F}\left( \hat{x}_1\right) -\bar{F}\left( \hat{x}_2\right) \right] . \end{aligned}$$

  Subsequently, substituting the equation above into the expression of $\hat{k}$, we have

  $$\begin{aligned} \hat{k}=cq^*-\{B/\Omega _2+\bar{F}\left( \hat{x}_3\right) -\theta \left[ \bar{F}\left( \hat{x}_1\right) -\bar{F}\left( \hat{x}_2\right) \right] \}w^*S^{-1}\left( \mathscr {A}_2\right) . \end{aligned}$$

  Taking the derivative of $\hat{k}$ with respect to b yields

  $$\begin{aligned} \frac{\partial \hat{k}}{\partial b}=S^{-1}\left( \mathscr {A}_2\right) \theta w^* \frac{\partial \bar{F}\left( \hat{x}_1\right) }{\partial b}=\theta f\left( \hat{x}_1\right) S^{-1}\left( \mathscr {A}_2\right) . \end{aligned}$$

  Thus, we can verify $\frac{\partial \hat{k}}{\partial b}>0$ because $S^{-1}\left( \mathscr {A}_2\right) >0$.

• (2) Taking the derivative of $\hat{k}$ with respect to d yields

  $$\begin{aligned} \frac{\partial \hat{k}}{\partial d}=-S^{-1}\left( \mathscr {A}_2\right) \theta w^* \frac{\partial \bar{F}\left( \hat{x}_2\right) }{\partial d}=-\theta f\left( \hat{x}_2\right) S^{-1}\left( \mathscr {A}_2\right) . \end{aligned}$$

  Thus, it is easy to verify that $\frac{\partial \hat{k}}{\partial d}<0$.

• (3) Taking the derivative of $\hat{k}$ with respect to $\theta$ yields

  $$\begin{aligned} \frac{\partial \hat{k}}{\partial \theta }=w^* \left[ \bar{F}\left( \hat{x}_1\right) -\bar{F}\left( \hat{x}_2\right) \right] S^{-1}\left( \mathscr {A}_2\right) . \end{aligned}$$

  Therefore, we can deduce that $\frac{\partial \hat{k}}{\partial \theta }>0$ because we have proved $\bar{F}\left( \hat{x}_1\right) -\bar{F}\left( \hat{x}_2\right) >0$.

1.8 Proof of Corollary 3

• (1) Based on our definition, we have $\eta =\frac{d{\Pi _s}^*}{dk}=\frac{\partial {\Pi _s}^*}{\partial q^*}\frac{dq^*}{dk}+\frac{\partial {\Pi _s}^*}{\partial k}$. According to the supplier’s FOC condition, the supplier’s optimal response $q^*$ satisfies $\frac{\partial {\Pi _s}^*}{\partial q^*}=0$. Thus, we have $\eta =\frac{\partial {\Pi _s}^*}{\partial k}=\frac{1}{1-r^*} \bar{F}\left( \hat{x}_3\right)$. Substituting Eq.(A.3) into the above formula, we have

  $$\begin{aligned} \eta =1-\frac{B}{\Omega _2 \left( 1-r^*\right) }+\frac{\theta }{1-r^*}\left[ F\left( \hat{x}_2\right) -F\left( \hat{x}_1\right) \right] . \end{aligned}$$

  Taking the derivative of $\eta$ with respect to b yields

  $$\begin{aligned} \eta =\frac{\partial \eta }{\partial b}=-\frac{\theta }{1-r^*}\frac{\partial \hat{x}_1}{\partial b} f\left( \hat{x}_1\right) =\frac{\theta }{w^*(1-r^*)}f\left( \hat{x}_1\right) >0. \end{aligned}$$

• (2) Taking the derivative of $\eta$ with respect to b yields

  $$\begin{aligned} \eta =\frac{\partial \eta }{\partial d}=\frac{\theta }{1-r^*}\frac{\partial \hat{x}_2}{\partial d} f\left( \hat{x}_2\right) =-\frac{\theta }{w^*(1-r^*)}f\left( \hat{x}_2\right) <0. \end{aligned}$$

• (3) Taking the derivative of $\eta$ with respect to b yields

  $$\begin{aligned} \eta =\frac{\partial \eta }{\partial \theta }=\frac{\left[ F\left( \hat{x}_2\right) -F\left( \hat{x}_1\right) \right] }{1-r^*}>0. \end{aligned}$$

1.9 Proof of Proposition 3

From the perspective of the supplier’s unit production cost and based on the supplier’s positive value condition in Eq. (A.5), we have $B \le w^* S^{-1}\{S(q^*)-\frac{{w^N}^*}{w^*} \left[ S\left( {q^N}^*\right) -S\left( \hat{x}_3^N\right) \right] \}$, which implies that $c \le \left[ (1-r^*) w^*S^{-1}\left( \mathscr {A}_2\right) +k\right] /q^*$. Therefore, if $\mathscr {A}_2 \ge 0$, we obtain the threshold

$$\begin{aligned} \hat{c}=\frac{(1-r^*) w^*S^{-1}\left( \mathscr {A}_2\right) +k}{q^*}, \end{aligned}$$

satisfying $\hat{c}q^*-k>0$. When $c\le \hat{c}$, the value of LCI for the supplier $\Delta \Pi _s \ge 0$. Otherwise, the value of LCI for the supplier $\Delta \Pi _s < 0$.

1.10 Proof of Corollary 4

• (1) Based on the proof of Corollary 2, we have $1-r^*{=}B/\Omega _2+\bar{F}\left( \hat{x}_3\right) -\theta \left[ \bar{F}\left( \hat{x}_1\right) {-}\bar{F}\left( \hat{x}_2\right) \right]$. Subsequently, substituting the equation above into the expression of $\hat{c}$, we have

  $$\begin{aligned} \hat{c}=\frac{\{B/\Omega _2+\bar{F}\left( \hat{x}_3\right) -\theta \left[ \bar{F}\left( \hat{x}_1\right) -\bar{F}\left( \hat{x}_2\right) \right] \}w^*S^{-1}\left( \mathscr {A}_2\right) +k}{q^*}. \end{aligned}$$

  Taking the derivative of $\hat{c}$ with respect to b yields

  $$\begin{aligned} \frac{\partial \hat{c}}{\partial b}=-\frac{\theta w^*}{q^*} S^{-1}\left( \mathscr {A}_2\right) \frac{\partial \bar{F}\left( \hat{x}_1\right) }{\partial b}=-\frac{\theta }{q^*}f\left( \hat{x}_1\right) S^{-1}\left( \mathscr {A}_2\right) . \end{aligned}$$

  Thus, we can verify $\frac{\partial \hat{c}}{\partial b}<0$ because $S^{-1}\left( \mathscr {A}_2\right) >0$.

• (2) Taking the derivative of $\hat{c}$ with respect to d yields

  $$\begin{aligned} \frac{\partial \hat{c}}{\partial d}=\frac{\theta w^*}{q^*} S^{-1}\left( \mathscr {A}_2\right) \frac{\partial \bar{F}\left( \hat{x}_2\right) }{\partial d}=\frac{\theta }{q^*}f\left( \hat{x}_2\right) S^{-1}\left( \mathscr {A}_2\right) . \end{aligned}$$

  Thus, it is easy to verify that $\frac{\partial \hat{c}}{\partial d}>0$.

• (3) Taking the derivative of $\hat{c}$ with respect to $\theta$ yields

  $$\begin{aligned} \frac{\partial \hat{c}}{\partial \theta }=\frac{-w^* \left[ \bar{F}\left( \hat{x}_1\right) -\bar{F}\left( \hat{x}_2\right) \right] S^{-1}\left( \mathscr {A}_2\right) }{q^*}. \end{aligned}$$

  Therefore, we can deduce that $\frac{\partial \hat{c}}{\partial \theta }<0$ because we have proved that $\bar{F}\left( \hat{x}_1\right) -\bar{F}\left( \hat{x}_2\right) >0$.