Competition among supply chains: the choice of financing strategy

Abstract

This article aims at studying the incentive for factoring in competing supply chains to examine the influences of the factoring decision in one supply chain on other chains. We consider multiple competing supply chains, each consisting of a manufacturer and a retailer. We firstly examine the issue under the special case of two supply chains, and then extend the model to multiple supply chains. The extended model is actually a bi-level non-cooperative game with multiple leaders and multiple followers, which we formulate as an Equilibrium Problem with Equilibrium Constraints. Numerical examples are presented to illustrate the rationality of the proposed model for the supply chains competition with financing choices.

Appendix

Proof of Proposition 1

According to the problem (5), it is easy to obtain:

• (1) If $ar-2[(a-c)-K(4-r^2)]\ge 0$, the optimal reaction for manufacturer i can be expressed as

  $$\begin{aligned} \begin{array}{rrr} w_i^{*(N,N)}(w^{(N,N)}_j)= \left\{ \begin{array}{lll} \frac{2(a+c)-r(a-w^{(N,N)}_j)}{4}, w^{(N,N)}_j\le \frac{ar-2[(a-c)-K(4-r^2)]}{r} \\ \frac{a(2-r)+rw^{(N,N)}_j-K(4-r^2)}{2}, w^{(N,N)}_j>\frac{ar-2[(a-c)-K(4-r^2)]}{r} \end{array} \right. \\ i,j=1,2, i\ne j \end{array} \end{aligned}$$

  If $w^{(N,N)}_j\le \frac{ar-2[(a-c)-K(4-r^2)]}{r}$, we say manufacturer j adopts a “high wholesale price” strategy, otherwise, adopts a “low wholesale price” strategy. Consequently, the optimal decisions and pay-off matrix for the two competing manufacturers are showed in Tables 4 and 5, respectively.

$\square$

Table 4 The optimal decisions for manufacturers

Table 5 The pay-off matrix for the manufacturers

For every $K\le \frac{2(a-c)}{(4 - r)(2+r)}$, it is easy to verify that $K [(a - c) - K (2 + r)]-\frac{K(2-r)[(4+r)(a-c)-4K(2+r)]}{8-r^2}\ge 0$. Consequently, the two manufacturers have incentive to increase their wholesale prices to maximize the pay-offs. Hence, if $ar-2[(a-c)-K(4-r^2)]\ge 0$, the equilibrium decisions for the two manufacturers are $w^{(N,N)}_i=a-(2+r)K$, $i=1,2$.

(2) If $ar-2[(a-c)-K(4-r^2)]< 0$, the optimal reaction for manufacturer i is

$$\begin{aligned} w_i^*(w_j)= \frac{a(2-r)+rw_j-K(4-r^2)}{2} \quad i, j=1,2, i\ne j \end{aligned}$$

Consequently the equilibrium decisions for the two manufacturers are $w^{*(N,N)}_i=a-(2+r)K$, $i=1,2$. Proposition 1 is proofed.

Specially, the proofs of Propositions 3 and 4 are similar to that of Proposition 1, and we omit it here.

Proof of Proposition 2

Based on the conclusions proposed in Proposition 1, the proposition can be easily obtained. $\square$

Proof of Proposition 5

Based on the conclusions proposed in Propositions 1, 2 and 4, the proposition can be obtained. $\square$

Proof of Theorem 1

• (1) According to Proposition 5, for every $\eta \le \bar{\eta }$, $\pi _1^{*(F,N)}\ge \pi _1^{*(N,N)}$ and $\pi _1^{*(F,F)}\ge \pi _1^{*(N,F)}$. Consequently, the retailers prefer the strategy F to N. Hence, under the wholesale price only, the equilibrium financing strategies of the two supply chains depend only on the incentive of the manufacturers to adopt factoring.

• (2) Given the financing strategy of each supply chain as N, we can conclude that $\varPi _1^{*(F,N)}\ge \varPi _1^{*(N,N)}$ and $\varPi _2^{*(F,F)}\ge \varPi _2^{*(F,N)}$ for every $\eta \le \bar{\eta }$ and $K\le K_2$. Consequently, if the initiated financing arrangement is (N, N), manufacturer 1 has incentive to propose factoring finance, resulting in a new financing arrangement of (F, N). Furthermore, manufacturer 2’s unique response is F, which causes the pair of the strategies yields (F, F). The process can be summarized as $(N,N)\rightarrow (F,N)$ or $(N,F)\rightarrow (F,F)$. Hence, (F, F) is the unique equilibrium financing strategy for every $\eta \le \bar{\eta }$ and $K\le K_2$.

• (3) For every $\eta \le \bar{\eta }$ and $K_4\le K\le K_2$, $\varPi _i^{*(F,F)}\le \varPi _i^{*(N,N)}$, hence (F, F) is the prison’s dilemma.

• (4) Similar to the above proofs, we can obtain the rest conclusions in Theorem 1.

$\square$

Proof of Theorem 2

The proof is similar to that of Theorem 1, therefore, we omit it here. $\square$

Proof of Theorem 3

By concavity and the maximum principle, for given $w_i$, we know that $q_i$ is a Nash equilibrium if and only if for each $i = 1, 2,\ldots ,m$

$$\begin{aligned} -(q_{i}-q^*_{i})^T \nabla _{q_i}\pi _i(q^*; w_i)\ge 0, \quad \forall q_{i}\in R^{n}_+ \end{aligned}$$

Thus, if $q^*$ is a Nash equilibrium for the retailers, then by concatenating these inequalities, it follows easily that $q^*$ must solve variational inequalities problem (16).

Conversely, for given w, if $q^*$ solves variational inequalities problem (16), for each $i = 1,2,\ldots ,m$, let q be the tuple whose j-th sub-vector is equal to $q^*_j$ for $j \not = i$ and $q_i$ is an arbitrary element of the set $R_+$. The variational inequalities problem (16) then becomes the above inequality. Consequently, Theorem 3 is proofed. $\square$