Online sellers’ financing strategies in an e-commerce supply chain: bank credit vs. e-commerce platform financing

We study a supply chain with two competitive online sellers, in which the seller who suffers from capital distress may borrow from a commercial bank or an e-commerce platform. We consider three different supply chain models depending on how many sellers suffer from capital distress. We characterize the sellers’ optimal production decisions and the lenders’ optimal interest rates, and further examine the sellers’ financing preferences. We demonstrate that the e-commerce platform will charge a zero interest rate to the financially constrained seller if the production cost is beyond a certain threshold. It is observed that the financially constrained seller’s financing preference is correlated with the capital status and financing choice of its rival. Specifically, if only one of the sellers is financially constrained, it is always optimal for the capital-constrained seller to borrow from the e-commerce platform. In contrast, if both sellers suffer from capital distress, the two sellers prefer e-commerce platform financing when the e-commerce platform’s commission rate is relatively high or when the commission ratio is low while the production cost is high. Furthermore, it is more profitable for the bank or e-commerce platform to provide financing service to both capital-constrained sellers.

The research is supported by the National Natural Science Foundation of China under Grant No. 717901117.

Authors and Affiliations

1. Institute of Economic Law, China University of Mining and Technology, Xuzhou, Jiangsu Province, 221116, People’s Republic of China

   Shilin Cai

2. School of Business Administration, Zhejiang University of Finance & Economics, Hangzhou, Zhejiang, 310018, People’s Republic of China

   Qiang Yan

Corresponding author

Correspondence to Qiang Yan.

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Proof of Lemma 1

For Problems (1) and (2), we show that \({\pi }_{{S}_{i}}^{NB}({q}_{i}^{NB})\) is concave in \({q}_{i}^{NB}\). According to the first-order condition, the optimal production quantities are \({q}_{1}^{NB*}=\) \(\frac{2a-2c-2ak-a\lambda +\lambda c+ak\lambda +\lambda c{r}_{{b}_{2}}^{NB}}{\left(2-\lambda \right)\left(\lambda +2\right)\left(1-k\right)}\) and \({q}_{2}^{NB*}=\frac{2a-2c-2ak-a\lambda +\lambda c-2c{r}_{{b}_{2}}^{NB}+ak\lambda }{\left(2-\lambda \right)\left(\lambda +2\right)\left(1-k\right)}\). Substituting \({q}_{2}^{NB*}\) into Eq. (3), we show that \({\pi }_{B}^{NB}({r}_{{b}_{2}}^{NB})\) is also a concave function with respect to \({r}_{{b}_{2}}^{NB}\). Therefore, the optimal interest rate is \({r}_{{b}_{2}}^{NB*}=\frac{\left(2-\lambda \right)[a\left(1-k\right)-c]}{4c}>0\). Substituting \({r}_{{b}_{2}}^{NB*}\) into \({q}_{1}^{NB*}\) and \({q}_{2}^{NB*}\), we further obtain \({q}_{1}^{NB*}=\frac{\left(\lambda +4\right)[a\left(1-k\right)-c]}{4\left(1-k\right)\left(\lambda +2\right)}\) and \({q}_{2}^{NB*}=\frac{a\left(1-k\right)-c}{2\left(1-k\right)\left(\lambda +2\right)}\).\(\square\)

Proof of Lemma 2

For Problems (4) and (5), it is straightforward that \({\pi }_{{S}_{i}}^{NE}({q}_{i}^{NE})\) is concave in \({q}_{i}^{NE}\). According to the first-order condition, the optimal production quantities are \({q}_{1}^{NE*}=\frac{2a-2c-2ak-a\lambda +\lambda c+ak\lambda +\lambda c{r}_{{p}_{2}}^{NE}}{\left(2-\lambda \right)\left(\lambda +2\right)\left(1-k\right)}\) and \({q}_{2}^{NE*}=\frac{2a-2c-2ak-a\lambda +\lambda c-2c{r}_{{p}_{2}}^{NE}+ak\lambda }{\left(2-\lambda \right)\left(\lambda +2\right)\left(1-k\right)}\). Substituting \({q}_{1}^{NE*}\) and \({q}_{2}^{NE*}\) into Eq. (6), we have \({r}_{{e}_{2}}^{NE*}=\) \(\frac{{\left(2-\lambda \right)}^{2}(2c-2a+4ak-a\lambda +\lambda c-2a{k}^{2}+ak\lambda +k\lambda c)}{2c\left(4k+k{\lambda }^{2}+2{\lambda }^{2}-8\right)}\) and \(\frac{{\partial }^{2}{\pi }_{E}^{NE}({r}_{{p}_{2}}^{NE})}{\partial {({r}_{{p}_{2}}^{NE})}^{2}}=\frac{2{w}^{2}\left(4k+k{\lambda }^{2}+ 2{\lambda }^{2}-8\right)}{{\left(2-\lambda \right)}^{2}{\left(\lambda +2\right)}^{2}{\left(k-1\right)}^{2}}<0\). Note that if \(c\ge \frac{a\left(1-k\right)\left(2+\lambda -2k\right)}{\lambda (1+k)+2}\), then \({r}_{{e}_{2}}^{NE*}\le 0\); if \(c<\frac{a\left(1-k\right)\left(2+\lambda -2k\right)}{\lambda (1+k)+2}\), then \({r}_{{e}_{2}}^{NE*}>0\). Therefore, if \(c\ge \frac{a\left(1-k\right)\left(2+\lambda -2k\right)}{\lambda (1+k)+2}\), then \({r}_{{e}_{2}}^{NE*}=0\); if \(c<\frac{a\left(1-k\right)\left(2+\lambda -2k\right)}{\lambda (1+k)+2}\), then \({r}_{{e}_{2}}^{NE*}=\frac{{\left(2-\lambda \right)}^{2}(2c-2a+4ak-a\lambda +\lambda c-2a{k}^{2}+ak\lambda +k\lambda c)}{2c\left(4k+k{\lambda }^{2}+2{\lambda }^{2}-8\right)}\). Substituting \({r}_{{e}_{2}}^{NE*}\) into \({q}_{1}^{NE*}\) and \({q}_{2}^{NE*}\), we obtain optimal production quantities.\(\square\)

Proof of Proposition 1

1. (i)

   \(\frac{\partial {r}_{{b}_{2}}^{NB*}}{\partial k}=\frac{a\left(\lambda -2\right)}{4c}<0\), \(\frac{\partial {q}_{1}^{NB*}}{\partial k}=-\frac{c\left(\lambda +4\right)}{4{\left(k-1\right)}^{2}\left(\lambda +2\right)}<0\) and \(\frac{\partial {q}_{2}^{NB*}}{\partial k}=-\frac{c}{2{\left(k-1\right)}^{2}\left(\lambda +2\right)}<0\);

2. (ii)

   If \(c<\frac{a\left(1-k\right)\left(2+\lambda -2k\right)}{\lambda (1+k)+2}\), then \(\frac{\partial {r}_{{e}_{2}}^{NE*}}{\partial k}=-\frac{{\left(\lambda -2\right)}^{2}\left(\begin{array}{c}24a-32ak+4a\lambda +12c\lambda +8a{k}^{2}-10a{\lambda }^{2}\\ -3a{\lambda }^{3}+2c{\lambda }^{2}-c{\lambda }^{3}+2a{k}^{2}{\lambda }^{2}+8ak{\lambda }^{2}+8c\end{array}\right)}{2c{\left(4k+k{\lambda }^{2}+2{\lambda }^{2}-8\right)}^{2}}<0\), \(\frac{\partial {q}_{1}^{NE*}}{\partial k}=-\frac{\begin{array}{c}64c+8a\lambda -64ck-8c\lambda -12a{\lambda }^{2}+2a{\lambda }^{3}+a{\lambda }^{4}+16{ck}^{2}-20c{\lambda }^{2}-2c{\lambda }^{3}+3c{\lambda }^{4} \\ -12{ak}^{2}{\lambda }^{2}+2{ak}^{2}{\lambda }^{3}+{ak}^{2}{\lambda }^{4}+8{ck}^{2}{\lambda }^{2}-2{ck}^{2}{\lambda }^{3}+{ck}^{2}{\lambda }^{4}-16ak\lambda +16ck\lambda \\ +24ak{\lambda }^{2}+8{ak}^{2}\lambda -4ak{\lambda }^{3}-2ak{\lambda }^{4}-8ck{\lambda }^{2}-8{ck}^{2}\lambda +4ck{\lambda }^{3}+2ck{\lambda }^{4}\end{array}}{2{\left(k-1\right)}^{2}{\left(4k+k{\lambda }^{2}+2{\lambda }^{2}-8\right)}^{2}}<0\) and \(\frac{\partial {q}_{2}^{NE*}}{\partial k}=\frac{\begin{array}{c}8a-16ak-12a\lambda +16ck+12c\lambda +8a{k}^{2}+2a{\lambda }^{2}+a{\lambda }^{3}-8c{k}^{2}-2c{\lambda }^{2}-c{\lambda }^{3}+2a{k}^{2}{\lambda }^{2}\\ +a{k}^{2}{\lambda }^{3}-2c{k}^{2}{\lambda }^{2}+24ak\lambda -8ck\lambda -4ak{\lambda }^{2}-12a{k}^{2}\lambda -2ak{\lambda }^{3}+4ck{\lambda }^{2}-2ck{\lambda }^{3}-8c\end{array}}{{\left(k-1\right)}^{2}{\left(4k+k{\lambda }^{2}+2{\lambda }^{2}-8\right)}^{2}}<0.\)

If \(c\ge \frac{a\left(1-k\right)\left(2+\lambda -2k\right)}{\lambda (1+k)+2}\), then \(\frac{\partial {r}_{{e}_{2}}^{NE*}}{\partial k}=0\) and \(\frac{\partial {q}_{1}^{NE*}}{\partial k}=\frac{\partial {q}_{2}^{NE*}}{\partial k}=-\frac{c}{{\left(k - 1\right)}^{2}\left(\lambda +2\right)}<0\).\(\square\)

Proof of Proposition 2

First, if \(c\ge \frac{a\left(1-k\right)\left(2+\lambda -2k\right)}{\lambda (1+k)+2}\), then \({r}_{{b}_{2}}^{NB*}-{r}_{{e}_{2}}^{NE*}=\frac{\left(2-\lambda \right)\left[a\left(1-k\right)-c\right]}{4w}>0\); if \(c<\) \(\frac{a\left(1-k\right)\left(2+\lambda -2k\right)}{\lambda (1+k)+2}\), then \({r}_{{b}_{2}}^{NB*}-{r}_{{e}_{2}}^{NE*}=-\frac{k\left(2-\lambda \right)\left(4a+4c-4ak-4a\lambda +4\lambda c-a{\lambda }^{2}-{\lambda }^{2}c+4ak\lambda +ak{\lambda }^{2}\right)}{4c\left(4k+k{\lambda }^{2}+2{\lambda }^{2}-8\right)}\) \(>0\). Therefore, \({r}_{{e}_{2}}^{NE*}<{r}_{{b}_{2}}^{NB*}\).

Second, if \(c\ge \frac{a\left(1-k\right)\left(2+\lambda -2k\right)}{\lambda (1+k)+2}\), then \({q}_{1}^{NB*}-{q}_{1}^{NE*}=\frac{\lambda \left[a\left(1-k\right)-c\right]}{4\left(1-k\right)\left(\lambda +2\right)}>0\); if \(c<\) \(\frac{a\left(1-k\right)\left(2+\lambda -2k\right)}{\lambda (1+k)+2}\), then \({q}_{1}^{NB*}-{q}_{1}^{NE*}=\frac{k\lambda \left(4a+4c-4ak-4a\lambda +4\lambda c-a{\lambda }^{2}-{\lambda }^{2}c+4ak\lambda +ak{\lambda }^{2}\right)}{4\left(\lambda +2\right)\left(k-1\right)\left(4k+k{\lambda }^{2}+2{\lambda }^{2}-8\right)}>0\). Therefore, \({q}_{1}^{NB*}>{q}_{1}^{NE*}\).

Third, if \(c\ge \frac{a\left(1-k\right)\left(2+\lambda -2k\right)}{\lambda (1+k)+2}\), then \({q}_{2}^{NB*}-{q}_{2}^{NE*}=-\frac{a\left(1-k\right)-c}{2\left(1-k\right)\left(\lambda +2\right)}<0\); if \(c<\) \(\frac{a\left(1-k\right)\left(2+\lambda -2k\right)}{\lambda (1+k)+2}\), then \({q}_{2}^{NB*}-{q}_{2}^{NE*}=\frac{k\left(4a+4c-4ak-4a\lambda +4\lambda c-a{\lambda }^{2}-{\lambda }^{2}c+4ak\lambda +ak{\lambda }^{2}\right)}{2\left(\lambda +2\right)\left(1-k\right)\left(4k+k{\lambda }^{2}+2{\lambda }^{2}-8\right)}<0\). Therefore, \({q}_{2}^{NB*}<{q}_{2}^{NE*}\).\(\square\)

Proof of Proposition 3

If \(c\ge \frac{a\left(1-k\right)\left(2+\lambda -2k\right)}{\lambda (1+k)+2}\), then \({\pi }_{{S}_{2}}^{NE*}-{\pi }_{{S}_{2}}^{NB*}=\frac{3{\left[a\left(1-k\right)-c\right]}^{2}}{{4\left(1-k\right)\left(\lambda +2\right)}^{2}}>0\) and \({\pi }_{{S}_{1}}^{NE*}-{\pi }_{{S}_{1}}^{NB*}=-\frac{\lambda {\left[a\left(1-k\right)-c\right]}^{2}\left(\lambda +8\right)}{{16\left(1-k\right)\left(\lambda +2\right)}^{2}}<0\); if \(c<\frac{a\left(1-k\right)\left(2+\lambda -2k\right)}{\lambda (1+k)+2}\), then \({\pi }_{{S}_{2}}^{NE*}-{\pi }_{{S}_{2}}^{NB*}>0\) and \({\pi }_{{S}_{1}}^{NE*}-\) \({\pi }_{{S}_{1}}^{NB*}<0\). Therefore, \({\pi }_{{S}_{2}}^{NE*}>{\pi }_{{S}_{2}}^{NB*}\) and \({\pi }_{{S}_{1}}^{NE*}<{\pi }_{{S}_{1}}^{NB*}\).\(\square\)

Proof of Proposition 4

1. (i)

   (i) \({\pi }_{{S}_{1}}^{NB*}-{\pi }_{{S}_{1}}^{NN*}=\frac{\lambda {\left(c-a+ak\right)}^{2}\left(\lambda +8\right)}{16{\left(\lambda +2\right)}^{2}\left(1-k\right)}>0\) and \({\pi }_{{S}_{2}}^{NB*}-{\pi }_{{S}_{2}}^{NN*}=\) \(\frac{3{\left(c-a+ak\right)}^{2}}{4\left(\mathrm{k}-1\right){\left(\lambda +2\right)}^{2}}<0\).

2. (ii)

   (ii) If \(c<\frac{a\left(1-k\right)\left(2+\lambda -2k\right)}{\lambda (1+k)+2}\), then \({\pi }_{{S}_{1}}^{NN*}-{\pi }_{{S}_{1}}^{NE*}=\frac{\lambda \left(2-\lambda \right)\left(\begin{array}{c}2a-2c-4ak+a\lambda -c\lambda \\ +2a{k}^{2}-ak\lambda -ck\lambda \end{array}\right)\left(\begin{array}{c}32a-32c-48ak+4a\lambda +16ck-4c\lambda +16a{k}^{2}\\ -8a{\lambda }^{2}-a{\lambda }^{3}+8c{\lambda }^{2}+c{\lambda }^{3}+2a{k}^{2}{\lambda }^{2}-8ak\lambda +\\ 6ak{\lambda }^{2}+4a{k}^{2}\lambda +ak{\lambda }^{3}+2ck{\lambda }^{2}+ck{\lambda }^{3}\end{array}\right)}{4{\left(\lambda +2\right)}^{2}\left(k-1\right){\left(4k+k{\lambda }^{2}+2{\lambda }^{2}-8\right)}^{2}}<0\); \({\pi }_{{S}_{2}}^{NN*}-{\pi }_{{S}_{2}}^{NE*}=\frac{\left(2-\lambda \right)\left(\begin{array}{c}12a-12c-16ak+8ck+4a{k}^{2}-3a{\lambda }^{2}+3c{\lambda }^{2}+\\ 2a{k}^{2}{\lambda }^{2}-2ak\lambda +2ck\lambda +ak{\lambda }^{2}+2a{k}^{2}\lambda +ck{\lambda }^{2}\end{array}\right)\left(\begin{array}{c}2a-2c-4ak+a\lambda -c\lambda \\ +2a{k}^{2}-ak\lambda -ck\lambda \end{array}\right)}{{\left(\lambda +2\right)}^{2}\left(1-k\right){\left(4k+k{\lambda }^{2}+2{\lambda }^{2}-8\right)}^{2}}>0\);

If \(c\ge \frac{a\left(1-k\right)\left(2+\lambda -2k\right)}{\lambda (1+k)+2}\), then \({\pi }_{{S}_{1}}^{NN*}-{\pi }_{{S}_{1}}^{NE*}={\pi }_{{S}_{2}}^{NN*}-{\pi }_{{S}_{2}}^{NE*}=0\);

Therefore, we have \({\pi }_{{S}_{1}}^{NE*}\ge {\pi }_{{S}_{1}}^{NN*}\) and \({\pi }_{{S}_{2}}^{NE*}\le {\pi }_{{S}_{2}}^{NN*}\). \(\square\)

Proof of Proposition 5

1. (1)

   First, \({\pi }_{B}^{NB*}=\frac{\left(2-\lambda \right){\left(c-a+ak\right)}^{2}}{8\left(1-k\right)\left(\lambda +2\right)}>{\pi }_{B}^{NE*}=0\).

   Second, if \(c<\frac{a\left(1-k\right)\left(2+\lambda -2k\right)}{\lambda \left(1+k\right)+2}\), then \({\pi }_{E}^{NE*}-{\pi }_{E}^{NB*}=\frac{{G}_{1}}{16{\left(\lambda +2\right)}^{2}{\left(k-1\right)}^{2}\left(4k+k{\lambda }^{2}+2{\lambda }^{2}-8\right)}\) and \({G}_{1}={a}^{2}{k}^{4}{\lambda }^{4}-8{a}^{2}{k}^{4}{\lambda }^{3}-8{a}^{2}{k}^{4}{\lambda }^{2}+32{a}^{2}{k}^{4}\lambda -48{a}^{2}{k}^{4}+16{a}^{2}{k}^{3}{\lambda }^{3}-64{c}^{2}\)

   $$-16{a}^{2}{k}^{3}{\lambda }^{2}-64{a}^{2}{k}^{3}\lambda +192{a}^{2}{k}^{3}-7{a}^{2}{k}^{2}{\lambda }^{4}-8{a}^{2}{k}^{2}{\lambda }^{3}+88{a}^{2}{k}^{2}{\lambda }^{2}-4{a}^{2}{\lambda }^{4}+32{a}^{2}{k}^{2}\lambda -304{a}^{2}{k}^{2}+10{a}^{2}k{\lambda }^{4}-96{a}^{2}k{\lambda }^{2}+224{a}^{2}k+32{a}^{2}{\lambda }^{2}-64{a}^{2}-4{c}^{2}{\lambda }^{4}$$
   $$+6ac{k}^{3}{\lambda }^{4}-48ac{k}^{3}{\lambda }^{2}-32ac{k}^{3}-2ac{k}^{2}{\lambda }^{4}-16ac{k}^{2}{\lambda }^{2}+224ac{k}^{2}+{c}^{2}{k}^{2}{\lambda }^{4}-12ack{\lambda }^{4}+128ack{\lambda }^{2}-320ack+8ac{\lambda }^{4}-64ac{\lambda }^{2}+128ac+8{c}^{2}{k}^{2}{\lambda }^{3}+32{c}^{2}{\lambda }^{2}$$

   \(-8{c}^{2}{k}^{2}{\lambda }^{2}-32{c}^{2}{k}^{2}\lambda -48{c}^{2}{k}^{2}+2{c}^{2}k{\lambda }^{4}-32{c}^{2}k{\lambda }^{2}+96{c}^{2}k<0\), which implies \({\pi }_{E}^{NE*}>{\pi }_{E}^{NB*}\). If \(c\ge \frac{a\left(1-k\right)\left(2+\lambda -2k\right)}{\lambda \left(1+k\right)+2}\), then \({\pi }_{E}^{NE*}-{\pi }_{E}^{NB*}=\frac{k\left(c-a+ak\right)\left(\begin{array}{c}4ak-4a+8a\lambda -8c\lambda -a{\lambda }^{2}\\ +5c{\lambda }^{2}-8ak\lambda +ak{\lambda }^{2}-12c\end{array}\right)}{16{\left(\lambda +2\right)}^{2}{\left(k-1\right)}^{2}}\) \(>0\). Therefore, \({\pi }_{E}^{NE*}>{\pi }_{E}^{NB*}\).

2. (2)

   First, \({\pi }_{B}^{NB*}=\frac{\left(2-\lambda \right){\left(c-a+ak\right)}^{2}}{8\left(1-k\right)\left(\lambda +2\right)}>{\pi }_{B}^{NN*}=0\).

   Second, if \(c<\frac{a\left(1-k\right)\left(2+\lambda -2k\right)}{\lambda (1+k)+2}\), then \({\pi }_{E}^{NE*}-{\pi }_{E}^{NN*}=-\frac{{\left(\lambda -2\right)}^{2}{\left(\begin{array}{c}2a-2c-4ak+a\lambda -c\lambda \\ +2a{k}^{2}-ak\lambda -ck\lambda \end{array}\right)}^{2}}{4{\left(\lambda +2\right)}^{2}{\left(k-1\right)}^{2}\left(4k+k{\lambda }^{2}+2{\lambda }^{2}-8\right)}\) \(>0\); if \(c\ge \frac{a\left(1-k\right)\left(2+\lambda -2k\right)}{\lambda (1+k)+2}\), then \({\pi }_{E}^{NE*}={\pi }_{E}^{NN*}\). Therefore, \({\pi }_{E}^{NE*}\ge {\pi }_{E}^{NN*}\).

3. (3)

   \({\pi }_{E}^{NB*}-{\pi }_{E}^{NN*}=\frac{k\left(a-ak-c\right){G}_{2}}{16{\left(\lambda +2\right)}^{2}{\left(k-1\right)}^{2}}\), where \({G}_{2}=4ak-12c-4a+8a\lambda -8c\lambda -a{\lambda }^{2}+5c{\lambda }^{2}-8ak\lambda +ak{\lambda }^{2}\). We show that \({G}_{2}\) decreases with \(c\) and \({G}_{2}{|}_{c=(1-k)a}\) \(<0\). Since \({G}_{2}{|}_{c=0}=a\left({\lambda }^{2}-8\lambda +4\right)\left(k-1\right)\), then \({G}_{2}{|}_{c=0}>0\) if \(\lambda >0.5359\), and \({G}_{2}{|}_{c=0}<0\) if \(\lambda <0.5359\). Therefore, when \(\lambda <0.5359\), we have \({G}_{2}<0\) and \({\pi }_{E}^{NB*}<{\pi }_{E}^{NN*}\); when \(\lambda \ge 0.5359\), there exists a unique \({c}_{0}\) such that \({G}_{2}\ge 0\) and \({\pi }_{E}^{NB*}\ge {\pi }_{E}^{NN*}\) if \(c\le {c}_{0}\), and \({G}_{2}<0\) and \({\pi }_{E}^{NB*}<{\pi }_{E}^{NN*}\) if \(c>{c}_{0}\). \(\square\)

Proof of Lemma 3

For Problem (7), it is straightforward that \({\pi }_{{S}_{i}}^{BB}({q}_{i}^{BB})\) is a concave function with respect to \({q}_{i}^{BB}\). According to the first-order condition, the optimal production quantities are \({q}_{i}^{BB}=\frac{2{kq}_{i}^{BB}-2{q}_{i}^{BB}+a-\lambda {q}_{3-i}^{BB}-ka+k\lambda {q}_{3-i}^{BB}}{1+{r}_{{b}_{i}}^{BB}}\). Therefore, we have \({q}_{i}^{BB*}=\frac{\lambda ka-2ka-\lambda a-2c+2a-2c{r}_{{b}_{3-i}}^{BB}+\lambda c+\lambda c{r}_{{b}_{i}}^{BB}}{(\lambda +2)(2-\lambda )(1-k)}\). Substituting \({q}_{i}^{BB*}\) into Problem (8), according to the first-order condition, we have \({r}_{{b}_{1}}^{BB*}={r}_{{b}_{2}}^{BB*}=\frac{\left(1-k\right)a-c}{2c}>0\). Since \(\frac{{\partial }^{2}{\pi }_{B}^{BB}}{\partial {({r}_{{b}_{1}}^{BB})}^{2}}{|}_{{r}_{{b}_{1}}^{BB*}}<0\) and \(\frac{{\partial }^{2}{\pi }_{B}^{BB}}{\partial {({r}_{{b}_{1}}^{BB})}^{2}}\frac{{\partial }^{2}{\pi }_{B}^{BB}}{\partial {({r}_{{b}_{2}}^{BB})}^{2}}-\frac{{\partial }^{2}{\pi }_{B}^{BB}}{\partial {r}_{{b}_{1}}^{BB}\partial {r}_{{b}_{2}}^{BB}}\frac{{\partial }^{2}{\pi }_{B}^{BB}}{\partial {r}_{{b}_{2}}^{BB}\partial {r}_{{b}_{1}}^{BB}}{|}_{({r}_{{b}_{1}}^{BB*}, {r}_{{b}_{2}}^{BB*})}>0\), then \(({r}_{{b}_{1}}^{BB*},{r}_{{b}_{2}}^{BB*})\) is the unique optimal solution. Further, we obtain \({q}_{1}^{BB*}\) and \({q}_{2}^{BB*}\). \(\square\)

Proof of Lemma 4

From Problem (9), we obtain \(\frac{\partial {\pi }_{{S}_{i}}^{EE}}{\partial {q}_{i}^{EE}}=\left(k-1\right)\left(2{q}_{i}^{EE}-a+\lambda {q}_{3-i}^{EE}\right)-c({r}_{{e}_{i}}^{EE}+1)\). According to the first-order condition, we have \({q}_{i}^{EE}=\frac{2{kq}_{i}^{EE}-2{q}_{i}^{EE}+a-\lambda {q}_{3-i}^{EE}-ka+k\lambda {q}_{3-i}^{EE}}{1+{r}_{{e}_{i}}^{EE}}\).

Therefore, \({q}_{i}^{EE*}=\frac{\lambda ka-2ka-\lambda a-2c+2a-2c{r}_{{e}_{3-i}}^{EE}+\lambda c+\lambda c{r}_{{e}_{i}}^{EE}}{(\lambda +2)(2-\lambda )(1-k)}\). Substituting \({q}_{i}^{EE*}\) into Problem (10), then \(\frac{\partial {\pi }_{E}^{EE}}{\partial {r}_{{e}_{i}}^{EE}}=\frac{c\left(\begin{array}{c}8a-8c-16ak-4a\lambda +4c\lambda -16c{r}_{{e}_{i}}^{EE}+8a{k}^{2}-2a{\lambda }^{2}+a{\lambda }^{3}+2c{\lambda }^{2}-\\ c{\lambda }^{3}+4c{\lambda }^{2}{r}_{{e}_{i}}^{EE}-2c{\lambda }^{3}{r}_{{e}_{3-i}}^{EE}+2a{k}^{2}{\lambda }^{2}+12ak\lambda -4ck\lambda +8ck{r}_{{e}_{i}}^{EE}+\\ 8c\lambda {r}_{{e}_{3-i}}^{EE}-8a{k}^{2}\lambda -ak{\lambda }^{3}+4ck{\lambda }^{2}-ck{\lambda }^{3}-8ck\lambda {r}_{{e}_{3-i}}^{EE}+2ck{\lambda }^{2}{r}_{{e}_{i}}^{EE}\end{array}\right)}{{\left(\lambda -2\right)}^{2}{\left(\lambda +2\right)}^{2}{\left(k-1\right)}^{2}}\). From the first-order condition, we have \({r}_{{e}_{i}}^{EE*}({r}_{{e}_{3-i}}^{EE})=\frac{\begin{array}{c}8c-8a+16ak+4a\lambda -4c\lambda -8a{k}^{2}+2a{\lambda }^{2}-a{\lambda }^{3}+c{\lambda }^{3}\\ +2c{\lambda }^{3}{r}_{{e}_{3-i}}^{EE}-2a{k}^{2}{\lambda }^{2}-12ak\lambda +4ck\lambda -8c\lambda {r}_{{e}_{3-i}}^{EE}\\ +8a{k}^{2}\lambda +ak{\lambda }^{3}-4ck{\lambda }^{2}+ck{\lambda }^{3}+8ck\lambda {r}_{{e}_{3-i}}^{EE}-2c{\lambda }^{2}\end{array}}{2c\left(4k+k{\lambda }^{2}+2{\lambda }^{2}-8\right)}\). Therefore, we obtain \({r}_{{e}_{1}}^{EE*}={r}_{{e}_{2}}^{EE*}=\frac{2{k}^{2}a-\lambda kc-\lambda ka-4ka+2a-\lambda c+\lambda a-2c}{2c\left(2+\lambda -k\right)}\). Since \(\frac{{\partial }^{2}{\pi }_{E}^{EE}}{\partial {\left({r}_{{e}_{1}}^{EE}\right)}^{2}}{|}_{{r}_{{e}_{1}}^{EE*}}<0\) and \(\frac{{\partial }^{2}{\pi }_{E}^{EE}}{\partial {\left({r}_{{e}_{1}}^{EE}\right)}^{2}}\frac{{\partial }^{2}{\pi }_{E}^{EE}}{\partial {\left({r}_{{e}_{2}}^{EE}\right)}^{2}}-\frac{{\partial }^{2}{\pi }_{E}^{EE}}{\partial {r}_{{e}_{1}}^{EE}\partial {r}_{{e}_{2}}^{EE}}\frac{{\partial }^{2}{\pi }_{E}^{EE}}{\partial {r}_{{e}_{2}}^{EE}\partial {r}_{{e}_{1}}^{EE}}{|}_{\left({r}_{{e}_{1}}^{EE*},{ r}_{{e}_{2}}^{EE*}\right)}>0\), then \(({r}_{{e}_{1}}^{EE*},{ r}_{{e}_{2}}^{EE*})\) is the unique optimal solution.

Let \(y=2{k}^{2}a-\lambda kc-\lambda ka-4ka+2a-\lambda c+\lambda a-2c\), we find that \(y\) decreases with \(c\), \(y{|}_{c=0}=a\left(1-k\right)\left(2+\lambda -2k\right)>0\) and \(y{|}_{c=(1-k)a}=ak\left(\lambda +2\right)\left(k-1\right)<0\). Therefore, there exists a unique \({c}_{1}\) such that \(y\le 0\) if \(c\ge {c}_{1}\), and \(y>0\) if \(c<{c}_{1}\). This implies that \({r}_{{e}_{1}}^{EE*}={r}_{{e}_{2}}^{EE*}=\frac{2{k}^{2}a-\lambda kc-\lambda ka-4ka+2a-\lambda c+\lambda a-2c}{2c\left(2+\lambda -k\right)}\) if \(c<{c}_{1}\), and \({r}_{{e}_{1}}^{EE*}={r}_{{e}_{2}}^{EE*}=0\) if \(c\ge {c}_{1}\). Furthermore, we obtain \({q}_{1}^{EE*}\) and \({q}_{2}^{EE*}\). \(\square\)

Proof of Lemma 5

Proof of Lemma 5 is similar to that of Lemma 2, and hence is omitted.\(\square\)

Proof of Proposition 6

Proof of Proposition 6 is similar to that of Proposition 1, and hence is omitted. \(\square\)

Proof of Proposition 7

For Part (i),

(1) If \(0\le c<{c}_{1}\), then \({r}_{{b}_{1}}^{BB*}-{r}_{{e}_{1}}^{EB*}=\frac{{z}_{1}}{2c\left(k{\lambda }^{4}-4k{\lambda }^{2}-32k-20{\lambda }^{2}+{\lambda }^{4}+64\right)}\), where \({z}_{1}=\) \(32ak+16a\lambda +32ck-16c\lambda -32{ak}^{2}+4a{\lambda }^{2}-4a{\lambda }^{3}-a{\lambda }^{4}-4c{\lambda }^{2}+4c{\lambda }^{3}+c{\lambda }^{4}-4a{k}^{2}{\lambda }^{2}-4a{k}^{2}{\lambda }^{3}+a{k}^{2}{\lambda }^{4}-64ak\lambda +16ck\lambda +48a{k}^{2}\lambda +8ak{\lambda }^{3}+ck{\lambda }^{4}-20ck{\lambda }^{2}+ 4ck{\lambda }^{3}\). Since \({z}_{1}\) is a monotone function with respect to \(c\), \({z}_{1}{|}_{c=0}>0\) and \({z}_{1}{|}_{c={c}_{1}}>0\), we obtain \({z}_{1}>0\) and \({r}_{{b}_{1}}^{BB*}>{r}_{{e}_{1}}^{EB*}\).

, where \({z}_{2}=32a\lambda -64ak-64ck-32c\lambda +96a{k}^{2}-32a{k}^{3}+24a{\lambda }^{2}-4a{\lambda }^{3}-a{\lambda }^{5}-6a{\lambda }^{4}+32c{k}^{2}-24c{\lambda }^{2} +4c{\lambda }^{3}+ 6c{\lambda }^{4}+c{\lambda }^{5}+24a{k}^{2}{\lambda }^{2}+8a{k}^{2}{\lambda }^{3}-16a{k}^{3}{\lambda }^{2}-4a{k}^{2}{\lambda }^{4}-a{k}^{2}{\lambda }^{5}+2a{k}^ {3}{\lambda }^{4}+8c{k}^{2}{\lambda }^{3}-2c{k}^{2}{\lambda }^{4}-c{k}^{2}{\lambda }^{5}-48ak\lambda -16ck\lambda -32ak{\lambda }^{2}+16a{k}^{2}\lambda -4ak{\lambda }^{3}+8ak{\lambda }^{4}+2ak{\lambda }^{5}+48ck{\lambda }^{2}+16c{k}^{2}\lambda +12ck{\lambda }^{3}-4ck{\lambda }^{4}\). We observe that \({z}_{2}\) decreases with \(c\), \({z}_{2}{|}_{c={c}_{1}}<0\) and \({z}_{2}{|}_{c=0}=a\left(k-1\right){z}_{3}\), where \({z}_{3}=2{k}^{2}{\lambda }^{4}-16{k}^{2}{\lambda }^{2}-32{k}^{2}-k{\lambda }^{5}-2k{\lambda }^{4}-32\lambda +8k{\lambda }^{3}+8k{\lambda }^{2}+16k\lambda +64k+{\lambda }^{5}+6{\lambda }^{4}+4{\lambda }^{3}-24{\lambda }^{2}\). Since \({z}_{3}\) increases with \(k\), \({z}_{3}{|}_{k=0}\) \(<0\) and \({z}_{3}{|}_{k=1}>0\), then there exists a unique \({k}_{0}\) such that \({z}_{3}>0\) and \({z}_{2}{|}_{c=0}<\) \(0\) if \(k>{k}_{0}\), and \({z}_{3}<0\) and \({z}_{2}{|}_{c=0}>0\) if \(k<{k}_{0}\).

Therefore, when \(k>{k}_{0}\), we have \({z}_{2}{|}_{c={c}_{1}}<0\) and \({z}_{2}{|}_{c=0}<0\), i.e., \({r}_{{b}_{1}}^{BE*}>{r}_{{e}_{1}}^{EE*}\); when \(k<{k}_{0}\), we have \({z}_{2}{|}_{c=0}>0\) and \({z}_{2}{|}_{c={c}_{1}}<0\). There exists a unique \({\overline{c}}_{12}\) such that \({r}_{{b}_{1}}^{BE*}>{r}_{{e}_{1}}^{EE*}\) if \(c>{\overline{c}}_{12}\), and \({r}_{{b}_{1}}^{BE*}\le {r}_{{e}_{1}}^{EE*}\) if \(c\le {\overline{c}}_{12}\).

(2) If \({c}_{1}\le c<{c}_{2}\), then \({r}_{{b}_{1}}^{BB*}-{r}_{{e}_{1}}^{EB*}=\frac{{z}_{1}}{2c\left(k{\lambda }^{4}-4k{\lambda }^{2}-32k-20{\lambda }^{2}+{\lambda }^{4}+64\right)}>0\) and \({r}_{{b}_{1}}^{BE*}\) \(-{r}_{{e}_{1}}^{EE*}=\frac{\left(2-\lambda \right)\left(2+\lambda \right)\left(\begin{array}{c}8a-8c+{\lambda }^{2}c-a{\lambda }^{2}+k{\lambda }^{2}c+a{\lambda }^{2}k-2\lambda kc\\ -2\lambda a+2\lambda ka+2\lambda c-12ka+4{k}^{2}a+4kc\end{array}\right)}{c\left({\lambda }^{4}+{\lambda }^{4}k-20{\lambda }^{2}-4k{\lambda }^{2}-32k+64\right)}>0\);

(3) If \({c}_{2}\le c<{c}_{3}\), then \({r}_{{b}_{1}}^{BB*}-{r}_{{e}_{1}}^{EB*}=\frac{\left(1-k\right)a-c}{2c}>0\) and \({r}_{{b}_{1}}^{BE*}-{r}_{{e}_{1}}^{EE*}=\frac{\left(2-\lambda \right)\left[\left(1-k\right)a-c\right]}{4c}>0\);

(4) If \(c\ge {c}_{3}\), then \({r}_{{b}_{1}}^{BB*}-{r}_{{e}_{1}}^{EB*}=\frac{\left(1-k\right)a-c}{2c}>0\) and \({r}_{{b}_{1}}^{BE*}-{r}_{{e}_{1}}^{EE*}=0\).

Similar to the proof of Part (i), we easily obtain Part (ii), Part (iii) and Part (iv). \(\square\)

Proof of Proposition 8

From seller 1’s perspective,

1. (1)

   \(0\le c<{c}_{1}\)

If seller 2 adopts bank credit, then \({\pi }_{{S}_{1}}^{EB*}-{\pi }_{{S}_{1}}^{BB*}=\frac{{F}_{1}{F}_{2}}{4{\left(\lambda +2\right)}^{2}\left(1-k\right){\left(\begin{array}{c}{\lambda }^{4}+{\lambda }^{4}k-20{\lambda }^{2}\\ -4k{\lambda }^{2}-32k+64\end{array}\right)}^{2}}\), where \({F}_{1}=128c-128a+160ak-16a\lambda -96kc+16\lambda c-32a{k}^{2}+36a{\lambda }^{2}+4a{\lambda }^{3}-a{\lambda }^{4}-36{\lambda }^{2}c-4{\lambda }^{3}c+{\lambda }^{4}c+4k{\lambda }^{2}c+4k{\lambda }^{3}c+k{\lambda }^{4}c-20a{k}^{2}{\lambda }^{2}-8ak{\lambda }^{3}+4a{k}^{2}{\lambda }^{3}+3a{k}^{2}{\lambda }^{4}+48ak\lambda -32k\lambda c-16ak{\lambda }^{2}-32a{k}^{2}\lambda -2ak{\lambda }^{4}<0\), \({F}_{2}=16\lambda c-16a\lambda -32kc-32ak+32a{k}^{2}-4a{\lambda }^{2}+4a{\lambda }^{3}+a{\lambda }^{4}+4{\lambda }^{2}c-4{\lambda }^{3}c-{\lambda }^{4}c+12k{\lambda }^{2}c+4k{\lambda }^{3}c-k{\lambda }^{4}c-12a{k}^{2}{\lambda }^{2}+4a{k}^{2}{\lambda }^{3}+a{k}^{2}{\lambda }^{4}+48ak\lambda -32k\lambda c+16ak{\lambda }^{2}-32a{k}^{2}\lambda -8ak{\lambda }^{3}-2ak{\lambda }^{4}<0\). Therefore, \({\pi }_{{S}_{1}}^{EB*}>{\pi }_{{S}_{1}}^{BB*}.\)

If seller 2 adopts e-commerce platform financing, then we have \({\pi }_{{S}_{1}}^{EE*}-{\pi }_{{S}_{1}}^{BE*}=\) \(\frac{{F}_{3}{F}_{4}}{4\left(k-1\right){\left(\lambda -k+2\right)}^{2} {\left({\lambda }^{4}+{\lambda }^{4}k-20{\lambda }^{2}-4k{\lambda }^{2}-32k+64\right)}^{2}}\), where \({F}_{3}=128c-128a+224ak-16a\lambda -160kc+16\lambda c-112a{k}^{2}+16a{k}^{3}+36a{\lambda }^{2}+4a{\lambda }^{3}-a{\lambda }^{4}+48{k}^{2}c+{\lambda }^{4}c-36{\lambda }^{2}c-4{\lambda }^{3}c+20k{\lambda }^{2}c-8{k}^{2}\lambda c-4k{\lambda }^{3}c+a{k}^{2}{\lambda }^{4}+8{k}^{2}{\lambda }^{2}c-{k}^{2}{\lambda }^{4}c+24ak\lambda +8k\lambda c-36ak{\lambda }^{2}-8a{k}^{2}\lambda -4ak{\lambda }^{3}<0\) and \({F}_{4}=32ak-16a\lambda +32kc+16\lambda c-48a{k}^{2}+16a{k}^{3}-4a{\lambda }^{2}+4a{\lambda }^{3}+a{\lambda }^{4}-16{k}^{2}c+4{\lambda }^{2}c-4{\lambda }^{3}c-{\lambda }^{4}c-12k{\lambda }^{2}c-8{k}^{2}\lambda c-4k{\lambda }^{3}c+8a{k}^{2}{\lambda }^{2}-a{k}^{2}{\lambda }^{4}+{k}^{2}{\lambda }^{4}c+24ak\lambda +8k\lambda c-4ak{\lambda }^{2}-8a{k}^{2}\lambda -4ak{\lambda }^{3}\). We observe that \({F}_{4}\) increases with \(c\), \({F}_{4}{|}_{c={c}_{1}}>0\) and \({F}_{4}{|}_{c=0}=a(k-1){y}_{2}\), where \({y}_{2}=16{k}^{2}-{\lambda }^{4}k+8k{\lambda }^{2}-8\lambda k-32k+4{\lambda }^{2}-4{\lambda }^{3}-{\lambda }^{4}+16\lambda\), \({y}_{2}\) decreases with \(k\), \({y}_{2}{|}_{k=0}=-{\lambda }^{4}+4{\lambda }^{2}-4{\lambda }^{3}+16\lambda >0\) and \({y}_{2}{|}_{k=1}=2\left({\lambda }^{2}+2\lambda -4\right)\left(2-{\lambda }^{2}\right)<0\). Hence, there exists a unique \({k}_{2}\) such that \({y}_{2}{|}_{k=0}>0\) and \({F}_{4}{|}_{c=0}<0\) if \(k<{k}_{2}\), and \({y}_{2}{|}_{k=0}\le 0\) and \({F}_{4}{|}_{c=0}\ge 0\) if \(k\ge {k}_{2}\), where \({k}_{2}\) holds \(16{k}^{2}-{\lambda }^{4}k+8k{\lambda }^{2}-{\lambda }^{4}\) \(-8\lambda k-32k+4{\lambda }^{2}-4{\lambda }^{3}+16\lambda =0\).

To sum up, when \(k\ge {k}_{2}\), then \({F}_{4}\ge 0\) and \({\pi }_{{S}_{1}}^{EE*}\ge {\pi }_{{S}_{1}}^{BE*}\); when \(k<{k}_{2}\), there exists a unique \({\overline{c}}_{1}\in (0,{c}_{1})\) such that \({F}_{4}<0\) and \({\pi }_{{S}_{1}}^{EE*}<{\pi }_{{S}_{1}}^{BE*}\) if \(c<{\overline{c}}_{1}\), and \({F}_{4}\ge 0\) and \({\pi }_{{S}_{1}}^{EE*}\ge {\pi }_{{S}_{1}}^{BE*}\) if \(c\ge {\overline{c}}_{1}\)

1. (2)

   \({c}_{1}\le c<{c}_{2}.\)

If seller 2 adopts bank credit, then \({\pi }_{{S}_{1}}^{EB*}-{\pi }_{{S}_{1}}^{BB*}=\frac{{F}_{1}{F}_{2}}{4{\left(\lambda +2\right)}^{2}\left(1-k\right){\left(\begin{array}{c}{\lambda }^{4}+{\lambda }^{4}k-20{\lambda }^{2}\\ -4k{\lambda }^{2}-32k+64\end{array}\right)}^{2}}\), \(>0\). If seller 2 adopts e-commerce platform financing, then \({\pi }_{{S}_{1}}^{EE*}-{\pi }_{{S}_{1}}^{BE*}=\frac{(2-\lambda ){F}_{5}{F}_{6}}{{\left(\lambda +2\right)}^{2}\left(1-k\right){\left({\lambda }^{4}+{\lambda }^{4}k-20{\lambda }^{2}-4k{\lambda }^{2}-32k+64\right)}^{2}}\), where \({F}_{5}=16a-16c-24ak+4a\lambda +8kc\) \(-4\lambda c+8a{k}^{2}-4a{\lambda }^{2}-a{\lambda }^{3}+4{\lambda }^{2}c+{\lambda }^{3}c+4k{\lambda }^{2}c+k{\lambda }^{3}c+2a{k}^{2}{\lambda }^{2}+a{k}^{2}{\lambda }^{3}-\) \(4ak\lambda +4k\lambda c+2ak{\lambda }^{2}>0\) and \({F}_{6}=96a-96c-144ak+8a\lambda +48kc-8\lambda c+48a{k}^{2}-28a{\lambda }^{2}-2a{\lambda }^{3}+a{\lambda }^{4}+28{\lambda }^{2}c+2{\lambda }^{3}c-{\lambda }^{4}c+4k{\lambda }^{2}c+2k{\lambda }^{3}c-k{\lambda }^{4}c+4a{k}^{2}{\lambda }^{2}-a{k}^{2}{\lambda }^{4}-16ak\lambda +24ak{\lambda }^{2}+8a{k}^{2}\lambda +2ak{\lambda }^{3}>0\). Hence, \({\pi }_{{S}_{1}}^{EE*}>{\pi }_{{S}_{1}}^{BE*}\). We conclude that seller 1 always prefers e-commerce platform financing.

1. (3)

   \({c}_{2}\le c<{c}_{3}\)

If seller 2 adopts bank credit, then \({\pi }_{{S}_{1}}^{EB*}-{\pi }_{{S}_{1}}^{BB*}=\frac{{[\left(1-k\right)a-c]}^{2}\left(\lambda +6\right)}{16\left(\lambda +2\right)\left(1-k\right)}>0\); if seller 2 adopts e-commerce platform financing, then \({\pi }_{{S}_{1}}^{EE*}-{\pi }_{{S}_{1}}^{BE*}=\frac{{3\left[\left(1-k\right)a-c\right]}^{2}}{4{\left(\lambda +2\right)}^{2}\left(1-k\right)}>0\). Therefore, seller 1 prefers e-commerce platform financing.

1. (4)

   \(c\ge {c}_{3}\)

If seller 2 adopts bank credit, then \({\pi }_{{S}_{1}}^{EB*}-{\pi }_{{S}_{1}}^{BB*}=\frac{{\left(\lambda +1\right)\left(\lambda +3\right)\left[\left(1-k\right)a-c\right]}^{2}}{4{\left(\lambda +2\right)}^{2}\left(1-k\right)}>0\); if seller 2 adopts e-commerce platform financing, we have \({\pi }_{{S}_{1}}^{EE*}-{\pi }_{{S}_{1}}^{BE*}=\frac{{[\left(1-k\right)a-c]}^{2}}{{\left(\lambda +2\right)}^{2}\left(1-k\right)}\) \(>0\). Therefore, it is optimal for seller 1 to adopt e-commerce platform financing.\(\square\)

Proof of Proposition 9

1. (i)

   First, if \(c<{c}_{2}\), then \({\pi }_{E}^{BB*}-{\pi }_{E}^{BE*}={M}_{1}-{M}_{5}<0\); if \({c}_{2}\le c<{c}_{3}\), then \({\pi }_{E}^{BB*}\) \(-{\pi }_{E}^{BE*}=\frac{k\left(c-a+ak\right)\left(2a+6c-2ak-a\lambda +5c\lambda +ak\lambda \right)}{16{\left(k-1\right)}^{2}\left(\lambda +2\right)}<0\); if \(c\ge {c}_{3}\), then \({\pi }_{E}^{BB*}-{\pi }_{E}^{BE*}=\) \(\frac{a\lambda \left(k-1\right)\left(20k-8\lambda +2k\lambda +k{\lambda }^{3}+8{\lambda }^{2}+2{\lambda }^{3}-32\right)}{4k+2\lambda -2k\lambda +k{\lambda }^{2}+{\lambda }^{2}-8}<0\). Therefore, \({\pi }_{E}^{BB*}<{\pi }_{E}^{BE*}\).

   Second, if \(c<{c}_{1}\), then \({\pi }_{E}^{BB*}-{\pi }_{E}^{EE*}=\frac{k\left(3ak-c-3a-a\lambda -c\lambda +ak\lambda \right)\left(c-a+ak\right)}{2{\left(k-1\right)}^{2}{\left(\lambda +2\right)}^{2}}-\frac{{\left(a-c\right)}^{2}}{2\left(2+\lambda -k\right)}\) \(<0\); if \(c\ge {c}_{1}\), then \({\pi }_{E}^{BB*}-{\pi }_{E}^{EE*}=\) \(\frac{k\left(c-a+ak\right)\left(a+3c-ak-a\lambda +3c\lambda +ak\lambda \right)}{2{\left(\lambda +2\right)}^{2}{\left(k-1\right)}^{2}}<0\). Therefore, \({\pi }_{E}^{BB*}<{\pi }_{E}^{EE*}\).

2. (ii)

   If \(c<{c}_{1}\), then \({\pi }_{E}^{EB*}-{\pi }_{E}^{EE*}={M}_{5}-\frac{{\left(a-c\right)}^{2}}{2\left(2+\lambda -k\right)}<0\);

   If \({c}_{1}\le c<{c}_{2}\), then \({\pi }_{E}^{EB*}-{\pi }_{E}^{EE*}=\frac{\left(2-\lambda \right){T}_{1}}{{\left(\lambda +2\right)}^{2}{\left(k-1\right)}^{2}{\left(32k+4k{\lambda }^{2}-k{\lambda }^{4}+20{\lambda }^{2}-{\lambda }^{4}-64\right)}^{2}}\), where \({T}_{1}=\frac{[{M}_{5}{\left(k-1\right)}^{2}{\left(\lambda +2\right)}^{2}-2k\left(ak-c-a-c\lambda \right)\left(c-a+ak\right)]{\left(\lambda +2\right)}^{2}{\left(k-1\right)}^{2}{\left(32k+4k{\lambda }^{2}-k{\lambda }^{4}+20{\lambda }^{2}-{\lambda }^{4}-64\right)}^{2} }{\left(2-\lambda \right){\left(k-1\right)}^{2}{\left(\lambda +2\right)}^{2}}\) and is a monotone function with respect to \(c\).

   When \(c={c}_{1}\), we have \({T}_{1}<0\);

   When \(c={c}_{2}\), we have \({T}_{1}=\frac{{a}^{2}{k}^{2}\left(2-\lambda \right){\left(\lambda +2\right)}^{2}{\left(k-1\right)}^{2}{\left(k{\lambda }^{4}-4k{\lambda }^{2}-32k-20{\lambda }^{2}+{\lambda }^{4}+64\right)}^{2}}{{\left(4k{\lambda }^{2}-4k\lambda -4\lambda +k{\lambda }^{3}+4{\lambda }^{2}+{\lambda }^{3}-16\right)}^{2}}{T}_{2}\), where \({T}_{2}=12k-4\lambda +4k\lambda -k{\lambda }^{2}+k{\lambda }^{3}+4{\lambda }^{2}+{\lambda }^{3}-16\) increases with \(k\), \({T}_{2}{|}_{k=0}<0\) and \({T}_{2}{|}_{k=1}=2{\lambda }^{3}+3{\lambda }^{2}-4\). We show that if \(\lambda >0.911\), then \({T}_{2}{|}_{k=1}>0\); if \(\lambda \le 0.911\), then \({T}_{2}{|}_{k=1}<0\). Therefore, when \(\lambda \le 0.911\), we have \({T}_{2}<0\); when \(\lambda >0.911\), we have \({T}_{2}{|}_{k=0}<0\) and \({T}_{2}{|}_{k=1}>0\), then there exists a unique \({k}_{22}=\frac{- {\lambda }^{3}-4{\lambda }^{2}+4\lambda +16}{{\lambda }^{3}-{\lambda }^{2}+4\lambda +12}\) such that if \(k<{k}_{22}\), then \({T}_{2}<0\); if \(k\ge {k}_{22}\), then \({T}_{2}\ge 0\).

   We conclude that: (1) when \(\lambda \le 0.911\), we have \({T}_{1}\le 0\) and \({\pi }_{E}^{EB*}\le {\pi }_{E}^{EE*}\); (2) when \(\lambda >0.911\) and \(k<{k}_{22}\), we have \({T}_{1}\le 0\) and \({\pi }_{E}^{EB*}\le {\pi }_{E}^{EE*}\); (3) when \(\lambda >0.911\) and \(k\ge {k}_{22}\), there exists a unique \({c}_{11}\in [{c}_{1},{c}_{2})\) such that if \(c<{c}_{11}\), then \({T}_{1}>0\) and \({\pi }_{E}^{EB*}>{\pi }_{E}^{EE*}\); if \(c\ge {c}_{11}\), then \({T}_{1}\le 0\) and \({\pi }_{E}^{EB*}\le {\pi }_{E}^{EE*}\).

   If \({c}_{2}\le c<{c}_{3}\), then \({\pi }_{E}^{EB*}-{\pi }_{E}^{EE*}=\frac{k\left(a-ak-c\right)}{16{\left(\lambda +2\right)}^{2}{\left(k-1\right)}^{2}}{T}_{3}\), where \({T}_{3}=4ak-12c-4a\) \(+8a\lambda -8c\lambda -a{\lambda }^{2}+5c{\lambda }^{2}-8ak\lambda +ak{\lambda }^{2}\). We show that \({T}_{3}\) decreases with \(c\), \({T}_{3}{|}_{c={c}_{3}}<0\) and \({T}_{3}{|}_{c={c}_{2}}=\frac{4a\left(2-\lambda \right)\left(\lambda +2\right)\left(1-k\right)}{4\lambda +4k\lambda -4k{\lambda }^{2}-k{\lambda }^{3}-4{\lambda }^{2}-{\lambda }^{3}+16}{T}_{2}\). Hence, when \(\lambda \le 0.911\), we have \({T}_{3}\le 0\) and \({\pi }_{E}^{EB*}\le {\pi }_{E}^{EE*}\); (2) when \(\lambda >0.911\) and \(k<{k}_{22}\), we have \({T}_{3}\le 0\) and \({\pi }_{E}^{EB*}\le {\pi }_{E}^{EE*}\); (3) when \(\lambda >0.911\) and \(k\ge {k}_{22}\), there exists a unique \({c}_{22}\in [{c}_{2},{c}_{3})\) such that if \(c<{c}_{22}\), then \({T}_{3}>0\) and \({\pi }_{E}^{EB*}>{\pi }_{E}^{EE*}\); if \(c\ge {c}_{22}\), then \({T}_{3}\le 0\) and \({\pi }_{E}^{EB*}\le {\pi }_{E}^{EE*}\).

   If \(c\ge {c}_{3}\), then \({\pi }_{E}^{EE*}-{\pi }_{E}^{EB*}=\frac{k\left(a-ak-c\right)\left(4a+4c-4ak-4a\lambda +4c\lambda -a{\lambda }^{2}-c{\lambda }^{2}+4ak\lambda +ak{\lambda }^{2}\right)}{4{\left(\lambda +2\right)}^{2}{\left(k-1\right)}^{2}}>0\).

   To sum up, if \(\lambda >0.911\), \(k\ge {k}_{22}\) and \(c\in ({c}_{1},{c}_{11})\cup ({c}_{2},{c}_{22})\), then \({\pi }_{E}^{EE*}<{\pi }_{E}^{EB*}\); otherwise, \({\pi }_{E}^{EE*}\ge {\pi }_{E}^{EB*}\).

3. (iii)

   First, if \(c<{c}_{2}\), then \({\pi }_{B}^{BB*}-{\pi }_{B}^{BE*}=\frac{{[\left(1-k\right)a-c]}^{2}}{2\left(1-k\right)\left(\lambda +2\right)}-{M}_{4}>0\); if \({c}_{2}\le c<{c}_{3}\), then \({\pi }_{B}^{BB*}-{\pi }_{B}^{BE*}=\frac{{\left[\left(1-k\right)a-c\right]}^{2}}{8\left(1-k\right)}>0\); if \(c\ge {c}_{3}\), then \({\pi }_{B}^{BB*}-{\pi }_{B}^{BE*}=\frac{{\left[\left(1-k\right)a-c\right]}^{2}}{2\left(1-k\right)\left(\lambda +2\right)}>0\). Therefore, \({\pi }_{B}^{BB*}>{\pi }_{B}^{BE*}\).

Second, since \({\pi }_{B}^{EE*}\) is always equal to 0, then we easily have \({\pi }_{B}^{BB*}>{\pi }_{B}^{BE*}>{\pi }_{B}^{EE*}\).\(=0.\square\)

Proof of Proposition 10

1. (i)

   \({\pi }_{{S}_{1}}^{NN*}-{\pi }_{{S}_{1}}^{BB*}={\pi }_{{S}_{2}}^{NN*}-{\pi }_{{S}_{2}}^{BB*}=\frac{3{\left(c-a+ak\right)}^{2}}{4\left(1-k\right){\left(\lambda +2\right)}^{2}}>0\);

   If \(c<{c}_{1}\), then \({\pi }_{{S}_{1}}^{NN*}-{\pi }_{{S}_{1}}^{EE*}={\pi }_{{S}_{2}}^{NN*}-{\pi }_{{S}_{2}}^{EE*}=\frac{\left(\begin{array}{c}6a-6c-8ak+3a\lambda +4ck\\ -3c\lambda +2a{k}^{2}-3ak\lambda +ck\lambda \end{array}\right)\left(\begin{array}{c}2a-2c-4ak+a\lambda -c\lambda \\ +2a{k}^{2}-ak\lambda -ck\lambda \end{array}\right)}{4{\left(\lambda +2\right)}^{2}\left(1-k\right){\left(\lambda -k+2\right)}^{2}}>0\);

   If \(c\ge {c}_{1}\), then \({\pi }_{{S}_{1}}^{NN*}-{\pi }_{{S}_{1}}^{EE*}={\pi }_{{S}_{2}}^{NN*}-{\pi }_{{S}_{2}}^{EE*}=0\).

   Therefore, \({\pi }_{{S}_{1}}^{NN*}-{\pi }_{{S}_{1}}^{EE*}={\pi }_{{S}_{2}}^{NN*}-{\pi }_{{S}_{2}}^{EE*}\ge 0\).

2. (ii)

   Proof of Part (ii) is similar to that of Part (i), and hence is omitted. \(\square\)

Proof of Proposition 11

1. (i)

   First, \({\pi }_{E}^{BB*}-{\pi }_{E}^{NN*}=\frac{k\left(c-a+ak\right)\left(a+3c-ak-a\lambda +3c\lambda +ak\lambda \right)}{2{\left(\lambda +2\right)}^{2}{\left(1-k\right)}^{2}}<0\);

   Second, if \(c<{c}_{1}\), then \({\pi }_{E}^{EE*}-{\pi }_{E}^{NN*}=\frac{{\left(2a-2c-4ak+a\lambda -c\lambda +2a{k}^{2}-ak\lambda -ck\lambda \right)}^{2}}{2{\left(\lambda +2\right)}^{2}{\left(1-k\right)}^{2}\left(2+\lambda -k\right)}>0\); if \(c\ge {c}_{1}\), then \({\pi }_{E}^{EE*}-{\pi }_{E}^{NN*}=0\). Therefore, \({\pi }_{E}^{EE*}\ge {\pi }_{E}^{NN*}\).

2. (ii)

   If \(c<{c}_{2}\), then \({\pi }_{E}^{BE*}-{\pi }_{E}^{NN*}=\frac{\left(2-\lambda \right){T}_{1}}{{\left(\lambda +2\right)}^{2}{\left(k-1\right)}^{2}{\left(32k+4k{\lambda }^{2}-k{\lambda }^{4}+20{\lambda }^{2}-{\lambda }^{4}-64\right)}^{2}}\).

   When \(c=0\), we have \({T}_{1}={a}^{2}{\left(k-1\right)}^{2}{T}_{4}\), where \({T}_{4}\) decreases with \(k\), \({T}_{4}{|}_{k=0}>0\) and \({T}_{4}{|}_{k=1}=-\left(2{\lambda }^{4}-9{\lambda }^{3}-22{\lambda }^{2}-4\lambda +8\right){\left({\lambda }^{2}+2\lambda -4\right)}^{2}\). We show that if \(>\) \(0.484\), then \({T}_{4}{|}_{k=1}>0\); if \(\lambda \le 0.484\), then \({T}_{4}{|}_{k=1}\le 0\). Therefore, when \(\lambda >0.484\), we have \({T}_{4}>0\); when \(\lambda \le 0.484\), there exist a unique \({k}_{11}\) such that if \(k\ge {k}_{11}\), then \({T}_{4}\le 0\); if \(k<{k}_{11}\), then \({T}_{4}>0\).

   When \(c={c}_{2}\), then \({T}_{1}=\frac{{a}^{2}{k}^{2}\left(2-\lambda \right){\left(\lambda +2\right)}^{2}{\left(k-1\right)}^{2}{\left(k{\lambda }^{4}-4k{\lambda }^{2}-32k-20{\lambda }^{2}+{\lambda }^{4}+64\right)}^{2}}{{\left(4k{\lambda }^{2}-4k\lambda -4\lambda +k{\lambda }^{3}+4{\lambda }^{2}+{\lambda }^{3}-16\right)}^{2}}{T}_{2}\). Hence, when \(\lambda \le 0.911\), we have \({T}_{2}<0\); when \(\lambda >0.911\), we have \({T}_{2}{|}_{k=0}<0\) and \({T}_{2}{|}_{k=1}>0\), then there exists a unique \({k}_{22}\) (\({k}_{22}<{k}_{11}\)) such that if \(k<{k}_{22}\), then \({T}_{2}<0\), and if \(k\ge {k}_{22}\), then \({T}_{2}\ge 0\).

   We conclude that: (1) When \(\lambda \le 0.484\), \(k>{k}_{11}\) and \(c<{c}_{2}\), then \({T}_{1}\le 0\) and \({\pi }_{E}^{BE*}\le {\pi }_{E}^{NN*}\); (2) when \(\lambda >0.484\), \(k>{k}_{22}\) and \(c<{c}_{2}\), then \({T}_{1}>0\) and \({\pi }_{E}^{BE*}>{\pi }_{E}^{NN*}\); (3) when \(\lambda \le 0.484\) and \(k\le {k}_{11}\) or when \(0.484<\lambda \le 0.911\) or \(\lambda >0.911\) and \(k\le {k}_{22}\), there exists a unique \({c}_{33}\in [0,{c}_{2})\) such that if \(c<{c}_{33}\), then \({T}_{1}>0\) and \({\pi }_{E}^{BE*}>{\pi }_{E}^{NN*}\); if \(c\ge {c}_{33}\), then \({T}_{1}\le 0\) and \({\pi }_{E}^{BE*}\le {\pi }_{E}^{NN*}\).

   If \({c}_{2}\le c<{c}_{3}\), then \({\pi }_{E}^{BE*}-{\pi }_{E}^{NN*}=\frac{k\left(a-ak-c\right){T}_{3}}{16{\left(\lambda +2\right)}^{2}{\left(k-1\right)}^{2}}\). Hence, when \(\lambda \le 0.911\), we have \({T}_{3}\le 0\) and \({\pi }_{E}^{EB*}\le {\pi }_{E}^{EE*}\); (2) when \(\lambda >0.911\) and \(k<{k}_{22}\), we have \({T}_{3}\le 0\) and \({\pi }_{E}^{EB*}\le {\pi }_{E}^{EE*}\); (3) when \(\lambda >0.911\) and \(k\ge {k}_{22}\), there exists a unique \({c}_{22}\in [{c}_{2},{c}_{3})\) such that if \(c<{c}_{22}\), then \({T}_{3}>0\) and \({\pi }_{E}^{BE*}>{\pi }_{E}^{EE*}\); if \(c\ge {c}_{22}\), then \({T}_{3}\le 0\) and \({\pi }_{E}^{BE*}\le {\pi }_{E}^{EE*}\).

   If \(c\ge {c}_{3}\), then \({\pi }_{E}^{BE*}-{\pi }_{E}^{NN*}=\frac{k\left(c-a+ak\right)\left(\begin{array}{c}4a+4c-4ak-4a\lambda +4c\lambda -a{\lambda }^{2}\\ -c{\lambda }^{2}+4ak\lambda +ak{\lambda }^{2}\end{array}\right)}{4{\left(\lambda + 2\right)}^{2}{\left(k-1\right)}^{2}}<0\).

   To sum up, if either of the following conditions holds: (1) \(\lambda >0.484\), \(k>{k}_{22}\) and \(c<{c}_{2}\); (2) \(\lambda \le 0.484\), \(k\le {k}_{11}\) and \(c<{c}_{33}\); (3) \(0.484<\lambda \le 0.911\), \(k\le 1\) and \(c<{c}_{33}\); (4) \(\lambda >0.911\), \(k\le {k}_{22}\) and \(c<{c}_{33}\); (5) \(\lambda >0.911\), \(k\ge {k}_{22}\) and \({c}_{2}\le c<{c}_{22}\), then we have \({\pi }_{E}^{BE*}>{\pi }_{E}^{NN*}\); otherwise, \({\pi }_{E}^{EB*}\le {\pi }_{E}^{EE*}\).

3. (iii)

   Since the bank’s profit is always equal to 0, we easily obtain \({\pi }_{B}^{BB*}>{\pi }_{B}^{NN*}=0\), \({\pi }_{B}^{EE*}={\pi }_{B}^{NN*}=0\) and \({\pi }_{B}^{BE*}\ge {\pi }_{B}^{NN*}=0\).\(\square\)

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