Coordinating loan strategies for supply chain financing with limited credit

Abstract

In this paper, we design a supply chain financing (SCF) system with a manufacturer, a retailer and a commercial bank where the retailer is capital constrained under demand uncertainties. We formulate a multi-level Stackelberg game in which the manufacturer acts as the leader and the bank as the sub-leader. Considering the bankruptcy risks of the retailer, we analyze the optimal credit line for the commercial bank, the optimal order for the retailer and the optimal wholesale price for the manufacturer, respectively. Comparing the benchmark scenarios of no capital constraint and constrained capital without financing, interdependencies between the operational and financial decisions are explored, as well as coordination analysis of the wholesale price contract with different credit lines. Finally, by conducting numerical studies, interactions between the operational and financial decisions and the impacts of credit lines on contract coordination are illustrated. The results validate our theoretical analysis. Our analysis suggests that a suitable financing scheme with limited credit would motivate the capital-constrained retailer to order more and the wholesale price contract with finite loans would achieve coordination in the SCF system.

Appendix

1.1 A.1 Proof of Lemma 1

Proof

From Eq. (1), If the retailer’s liquid assets from the sales at the end of the season can cover its loan obligations, i.e., $p\min (q,x)+(k_r +L(\theta )-wq)\ge L(\theta )(1+R_r ),$ then the retailer has no bankruptcy risk. Hence, when $x\le q$, the minimal realized demand $\hat{x}(q)={(wq-K_r )(1+\theta R_r )} / p$ can be easily obtained from the inequality of $px-(wq-k_r )(1+\theta R_r )\ge 0$. $\square$

1.2 A.2 Proof of Proposition 1

Proof

First, we solve the unconstrained optimization problem of Eq. (1). For notational convenience, let $\Omega =w(1+\theta R_r ) / p$ From Eq. (1), the retailer’s expected profit can be rewritten as $\pi _r =p \int _{{\hat{x}}}^{q} {\bar{F}(\hat{x})\mathrm{d}x}$. Taking the first-order and second-order derivative of $\pi _r$ with respect to $q$, it follows that ${\mathrm{d}\pi _r}/{\mathrm{d}q}=p\bar{F}(q)-p\bar{F}(\hat{x})\cdot {\mathrm{d}\hat{x}}/{\mathrm{d}q}=p({\bar{F}(q)-\Omega \bar{F}(\hat{x})})$ and ${\mathrm{d}^2\pi _{r} }/{\mathrm{d}q^2}=p({-f(q)+\Omega f(\hat{x})\cdot {\mathrm{d}\hat{x}}/{\mathrm{d}q}})=p({-f(q)+\Omega ^2f(\hat{x})})$. As $p>w(1+R_r)$ and $0<\theta <1$, it is obvious that $\Omega <1$ and thus $\Omega ^2<1$. If the distribution of demand is IFR, the inequality of $\Omega ^2f(\hat{x})-f(q)<0$ holds for $\hat{x}<q$, i.e., $\mathrm{d}^2{\pi _r }/{\mathrm{d}q^2<0}$. Hence, from $\mathrm{d}\pi _r {(q^*)}/{dq=0}$ we have $q^*=\bar{F}^{-1}({\Omega \bar{F}(\hat{x})})$.

Next, we consider the constrained optimization problem of Eqs. (1) and (2). Using Lagrangian relaxation method, we reformulate Eqs. (1) and (2) as $\max \nolimits _q L(q,\lambda )=\pi _r (q)+\lambda (wq-K_r)$. Then the optimal order quantity $q^{*}$ and Lagrangian multiplier $\lambda ^{*}$ satisfies the complementary slackness conditions such that $\mathrm{d}L{(q^*,\lambda ^*)} / {{\mathrm{d}q=0}} ,\,\lambda ^{*} (wq^{*}-K_r )=0,\,\,\lambda ^{*} \ge 0$, and $wq^*-K_r \ge 0$. Differentiating Lagrangian function $L(q,\lambda )$ with respect to $q$, we have $\mathrm{d}L{(q,\lambda )}/{{\mathrm{d}q=\mathrm{d}\pi _r}} {(q)}/{{\mathrm{d}q}} +w\lambda =p( {\bar{F}(q)-\Omega \bar{F}(\hat{x})})+w\lambda$. If $\lambda ^*>0$, $q^*$ satisfies $wq^*-K_r =0$. Substituting $q^*={K_r }/w$ into the first-order optimality condition of $\mathrm{d}L{(q^*,\lambda ^*)} / {\mathrm{d}q=0}$, the optimal $\lambda ^*=p{[{\Omega \bar{F}(\hat{x})-\bar{F}( {{K_r } / w})} ]} /w$, which implies $\lambda ^*\le 0$ since $\Omega \bar{F}(\hat{x})=\bar{F}(q)\le \bar{F}( {{K_r } / w})$. Hence, the optimal solution in this case should be disregarded because of the contradiction of $\lambda ^*$. Alternatively, in the case of non-binding constraint where $\lambda ^*=0$, the condition $\mathrm{d}L{(q^*,\lambda ^*)}/{\mathrm{d}q=0}$ can be transformed as $\mathrm{d}\pi _r {(q^*)}/{\mathrm{d}q^*=0}$, which is the same with the unconstrained case. Thus, the optimal solution $q^*=\bar{F}^{-1}( {\Omega \bar{F}(\hat{x})})$ is unique. $\square$

1.3 A.3 Proof of Lemma 2

Proof

Applying the Implicit Function Theorem of $\bar{F}(q^*)=\Omega \bar{F}( {\hat{x}(q^*)})$ from Proposition 1 yields $-f(q^*)\cdot {\mathrm{d}q^*}/{\mathrm{d}\theta } =\bar{F}( {\hat{x}(q^*)})\cdot {\mathrm{d}\Omega }/{ {\mathrm{d}\theta -\Omega f( {\hat{x}(q^*)})\cdot {\mathrm{d}( {\hat{x}(q^*)})} / }} {\mathrm{d}\theta }$. As ${\mathrm{d}( {\hat{x}(q^*)})}/{\mathrm{d}\theta } \quad {=( {R_r B(q^*)+w(1+\theta R_r )\cdot \mathrm{d}q^*/{\mathrm{d}\theta }})} / p$ and ${\mathrm{d}\Omega }/{\mathrm{d}\theta }={wR_r } / p$, the first-order derivate of $\pi _b$ with respect to $\theta$ can be obtained as ${\mathrm{d}q^*}/{\mathrm{d}}\theta$ = ${wR_r {[{\bar{F}(\hat{x})-(\hat{x})f(\hat{x})}]} / p[{\Omega ^{2} f(\hat{x})-f(q^*)}]}.$ If the demand distribution function is IFR, then $\hat{x}<q^*$ makes it easy to observe ${\hat{x}f(\hat{x})} / {\bar{F}(\hat{x}){<\Omega q{^*}f(q^*)} / {\bar{F}(q^*)\le 1}}$. Therefore, the inequality of $\bar{F}(\hat{x})-\hat{x}f(\hat{x})>0$ holds. Moreover, the inequality of $\Omega ^2f(\hat{x})-f(q)<0$ has been proved in Proposition 1, so the inequality of ${\mathrm{d}q^*(\theta )} / {\mathrm{d}\theta <0}$ holds. $\square$

1.4 A.4 Proof of Proposition 2

Proof

From Eq. (3), the bank’s expected profit can be expressed as $\pi _b \!=\!p\int _0^{\hat{x}} {\bar{F}(x)\mathrm{d}x\!-\!p\hat{x}\!+\!\theta ( {wq-k_r })( {R_r -R_f })}$. The first-order condition and the second-order condition of $\pi _b$ for $\theta$ can be given as ${\mathrm{d}\pi _b } / \mathrm{d}\theta =( {wq-K_r +\theta w\cdot {\mathrm{d}q} / {\mathrm{d}\theta }}) ( {R_r \bar{F}(\hat{x})-R_f })-wF(\hat{x})\cdot {\mathrm{d}q}/{\mathrm{d}\theta }$ and ${\mathrm{d}^2\pi _b }/\mathrm{d}\theta ^2=2w( {R_r \bar{F}(\hat{x})-R_f })\cdot {\mathrm{d}q}/\mathrm{d}\theta +w[{\theta ({R_r \bar{F}(\hat{x})-R_f })-F(\hat{x})}]\cdot {\mathrm{d}^{2}q}/{\mathrm{d}\theta ^{2}}-pf(\hat{x})\cdot ({{\mathrm{d}\hat{x}}/{\mathrm{d}\theta ^2}})$, respectively. Meanwhile, from Lemma 2, we can deduce that ${\mathrm{d}^2q^*}/{\mathrm{d}\theta ^2}=\,\,-2wR_r f(\hat{x})( {\Omega ^2f(\hat{x})-f(q^*)})\cdot {\mathrm{d}\hat{x}}/\mathrm{d}\theta +\Omega ( {\bar{F}(\hat{x})-\hat{x}f(\hat{x})})\cdot {\mathrm{d}\Omega }/{\mathrm{d}\theta } / {p[{\Omega ^2f(\hat{x})-f(q^*)} ]^2}$. Since ${\mathrm{d}\hat{x}}/{\mathrm{d}\theta }=(wq-K_r ){R_r } / {p+\Omega \cdot {\mathrm{d}q}/{\mathrm{d}\theta }}<0$ and ${\mathrm{d}\Omega }/{\mathrm{d}\theta >0}$, as well as the inequalities of $\bar{F}(\hat{x})-\hat{x}f(\hat{x})>0$ and $\Omega ^2f(\hat{x})-f(q)<0$ proved in Lemma 2, the inequality of ${\mathrm{d}^2q^*}/{\mathrm{d}\theta ^2<0}$ holds obviously. Substituting ${\mathrm{d}^2q^*}/{\mathrm{d}\theta ^2<0}$ and ${\mathrm{d}q^*} /{\mathrm{d}\theta }<0$, proved in Lemma 2, into $\mathrm{d}^2{\pi _b}/{\mathrm{d}\theta ^2}$, the inequality of ${\mathrm{d}^2\pi _b}/{\mathrm{d}\theta ^2}<0$ holds if ${\bar{F}(\hat{x})\ge R_f }/{R_r}$. Therefore, with ${\mathrm{d}\pi _b (\theta ^*)}/{\mathrm{d}\theta =0}$, the optimal $\theta ^*$ can be obtained. $\square$

1.5 A.5 Proof of Proposition 3

Proof

From Eq. (4), taking the first-order derivative of $\pi _m$ with respect to $w$, we have $\mathrm{d}\pi _m /\mathrm{d}w=\partial \pi _m /\partial q^*\cdot \mathrm{d}q^*/\mathrm{d}w+\partial \pi _m /\partial w=({w-c})\cdot \,\mathrm{d}q^*/\mathrm{d}w+q^*\cdot$ Hence, let $\mathrm{d}\pi _m /\mathrm{d}w=0$, the optimal wholesale price can be obtained as $w^*=c-\frac{q^*}{\mathrm{d}q^*/\mathrm{d}w}\cdot$

According to Proposition 1, applying the Implicit Function Theorem on $\bar{F}( {q^*})\!=\!\Omega \bar{F}( {\hat{x}( {q^*})})$ yields $\mathrm{d}q^*/\!\mathrm{d}w\!=\!-[{\bar{F}( {\hat{x}( {q^*})})\!\cdot \!\mathrm{d}\Omega \!/\mathrm{d}w\!-\!\Omega f( {\hat{x}( {q^*})})\!\cdot \! \mathrm{d}( {\hat{x}({q^*})})\!/\mathrm{d}w}]/\!f( {q^*})$. Moreover, the first-order derivate of $\hat{x}( {q^*})$ and $\Omega$ for $w$ are $\mathrm{d}({\hat{x}( {q^*})})/\mathrm{d}w=({1+\theta R_r })( {q^*\,+w\cdot \mathrm{d}q^*/\mathrm{d}w})/p$ and $\mathrm{d}\Omega /\mathrm{d}w=\Omega /w$, respectively; substitute them into $\mathrm{d}q^*/\mathrm{d}w$, and thus $\mathrm{d}q^*/\mathrm{d}w=\Omega [{\bar{F}( {\hat{x}})-\Omega q^*\,f( {\hat{x}})}]/w[{\Omega ^2\,f( {\hat{x}})-f( {q^*})}]$. As $\bar{F}( {\hat{x}})-\Omega q^*\,f( {\hat{x}})=\bar{F}( {\hat{x}})( {1-\Omega q^*\,f( {\hat{x}})/\bar{F}( {\hat{x}})})$ and $0<\Omega <1$, the inequality of $\Omega q^*\,f( {\hat{x}})/\bar{F}( {\hat{x}})<\Omega q^*\,f({q^*})/\bar{F}( {q^*})<1$ holds for the IFR demand distribution and $\hat{x}<q$. Meanwhile, because $\Omega ^2f( {\hat{x}})-f( q)<0$ proved in Proposition 1, the inequality of $\mathrm{d}q^*/\mathrm{d}w<0$ holds.

Next, we aim to show that the second-order derivative is negative, i.e., $\mathrm{d}^2\pi _m/\mathrm{d}w^2<0$. Taking the second-order derivative of $\pi _m$with respect to $w$ we have $\mathrm{d}^2\pi _m/\mathrm{d}w^2=2dq^*/\mathrm{d}w+( {w-c})\cdot \,\mathrm{d}^2\,q^*/\mathrm{d}w^2$. Recall that $\mathrm{d}q^*/\mathrm{d}w<0$ proved above and the assumption of $w\ge c$. Hence, we only need to prove $\mathrm{d}^2{q_{i}^{*}}/\mathrm{d}w_{i}^{2} <0$, where $\mathrm{d}^2q^*/\mathrm{d}w^2=-2\Omega ^2f( {\hat{x}})( {2\bar{F}( {q^*})-\Omega ^2q^*f( {\hat{x}})q^*f( {q^*})})/w^2( {\Omega ^2f( {\hat{x}})-f( {q^*})})^2$. Recall that $\bar{F}( {q^*})\ge \Omega ^2q^*f( {\hat{x}})$ and $\bar{F}( {q^*})\ge q^*f( {q^*})$, it is obvious that $\mathrm{d}^2q^*/\mathrm{d}w^2<0$. Hence, the second-order condition of $\pi _m$ for $w$ satisfies $\mathrm{d}^2\pi _m/\mathrm{d}w^2=2dq/\mathrm{d}w+( {w-c})\cdot \,\mathrm{d}^2q/\mathrm{d}w^2<0$. Therefore, from the first-order condition of $\mathrm{d}\pi _m ( {w^*})/\mathrm{d}w=q+\Omega ( {w^*-c})[{\bar{F}( {\hat{x}})-\Omega qf( {\hat{x}})}]/w^*[{\Omega ^2f( {\hat{x}})-f( q)}]=0$, the optimal $w^*$ can be obtained. $\square$

1.6 A.6 Proof of Proposition 4

Proof

From Proposition 1, we have $q^*=\bar{F}^{-1}( {\Omega \bar{F}( {\hat{x}})})$. It is obvious that the inequality of $q^{N^{*} }\ge q^{NB^{*}}$holds if $\bar{F}^{-1}( {\Omega \bar{F}( {\hat{x}})})\ge \bar{F}^{-1}( {\Omega ^N})$, which implies that $\Omega \bar{F}( {\hat{x}})\le \Omega ^N$ since $F(x)$ is strictly increasing and follows IFR. Hence, substituting $\Omega$ and $\Omega ^N$ into it we transform the inequality of $\Omega \bar{F}( {\hat{x}})\le \Omega ^N$ into $\bar{F}( {\hat{x}})\le ( {1+R_f })/( {1+\theta R_r })$, which implies that $\theta \le \bar{\theta }=( {{F}( {\hat{x}})+R_f })/R_r \bar{F}( {\hat{x}})$.

Moreover, according to the optimality of $\theta ^*$ given in Proposition 2, we have the inequality of ${\theta ^{*}} \ge \underline{\theta }={F}( {\hat{x}})/( {R_r \bar{F}( {\hat{x}})-R_f })>0$. Therefore, when $0<\underline{\theta }<\theta \le \bar{\theta }<1$, the inequality of $q^*\ge q^{N^{*}}$ holds. Moreover, it is obvious that $q^{N^{*} }\ge q^{\textit{NB}^{*}}$. Hence, the inequality of $q^*\ge q^{N^{*} }\ge q^{\textit{NB}^{*}}$ could be valid. $\square$

1.7 A.7 Proof of Proposition 5

Proof.

According to the above analysis, it is obvious that the coordination condition of $q^*( {\hat{w},\hat{\theta }})=q_{s}^{*} ( {\hat{w},\hat{\theta }})$ implies that $\bar{F}( {q^*( {\hat{w},\hat{\theta }})})=\bar{F}( {q_{s}^{*} ( {\hat{w},\hat{\theta }})})$, i.e., $\hat{w}( {1+\hat{\theta }R_r })\bar{F}( {\hat{x}})=( {\hat{\theta }\hat{w}R_f +c})\bar{F}( {\hat{x}_{s}})$. Therefore, if the credit line in the wholesale price contract with a finite loan satisfies the condition of $\hat{\theta }={( {\hat{w}\bar{F}(\hat{x})-c\bar{F}(\hat{x}_s )})} / {\hat{w}( {R_f \bar{F}(\hat{x}_s )-R_r \bar{F}(\hat{x})})}$, the SCF system can achieve coordination. Moreover, to guarantee the credit line $\hat{\theta }\in (0,1)$, the inequality of $1<{\hat{w}(1+R_r )} / {(\hat{w}R_f +c)}<{\bar{F}(\hat{x}_s )} / {\bar{F}(\hat{x})}$ holds for $\hat{w}>c$. $\square$

1.8 A.8 Proof of Proposition 6

Proof.

Similar to the proof of Proposition 5, if a coordinating wholesale price $\hat{w}^I$ with infinite loans exists, the necessary condition of $q_{\theta =1}^{*} ( {\hat{w}^I})=q_S^{I^{*} } ( {\hat{w}^I})$ for coordinating the SCF system should be satisfied. It is also implies that $\bar{F}( {q_{\theta =1}^{*} ( {\hat{w}^I})})=\bar{F}( {q_S^{I^{*} } ( {\hat{w}^I})})$, i.e.,$\hat{w}^I=c\bar{F}( {\hat{x}_s })/( {( {1+R_r })\bar{F}( {\hat{x}})-R_f \bar{F}( {\hat{x}_s })})$. However, since $R_r \ge R_f$ and $\bar{F}( {\hat{x}_{S}^{I} ( {q_{S}^{I^{*}} })})<\bar{F}( {\hat{x}( {q_{\theta =1}^*})})$ with the IFR demand distribution, the ratio of $\bar{F}( {\hat{x}_S })/( {( {1+R_r })\bar{F}( {\hat{x}})-R_f \bar{F}( {\hat{x}_S })})$ is less than 1, i.e., $\hat{w}^I<c$. Hence it’s straightforward to find that $q_{{\theta = 1}}^{*} (\hat{w}^{I}) = q_{s}^{{I^{*}}} (\hat{w}^{I})$ only if $\hat{w}^{I} <c.$ $\square$