#### 5.1. Two-period Bank Financing Problem

Recall our single-period static financing model under bank financing with retailer’s optimal ordering strategy and supplier’s optimal wholesale price, this subsection considers the two-period case under bank financing to analyze how to maximize the terminal cumulative profits at the end of the second periods with the perspective of the retailer and the supplier, respectively.

According to Equation (3), we get the retailer’s profit in the second period under the two-period bank financing scenario by replacing parameter

t with 2 and general interest rate r with

${r}_{b}$:

According to the conclusion of one-period financing model, if only the second period is discussed, i.e., t = 2 and

${x}_{1}$ is fixed, it can be regarded as a one-period problem, which can be obtained from Theorem 1:

Let denote

${l}_{2}^{b}$ to represent the retailer’s base-stock level in the second period under bank financing, which is composed of the retailer’s initial inventory at the beginning of the second period

${x}_{1}$ and his optimal ordering quantity in the second period

${q}_{2}^{b}$. After simplification of Equation (53), we have:

The retailer’s optimal order quantity satisfies Equation (55):

According to Bellman’s principle of optimality mentioned by Kogan [

36], no matter what the past state and decision of the process are, the remaining decision must constitute the optimal strategy for the state formed by the previous decision. Therefore, when considering the retailer’s decision in the first period, the decision in the second period can reach the optimal order quantity

${q}_{2}^{b}$.

Since the retailer’s cash flow and inventory are uncertain at the beginning of the second period, it is necessary to discuss the retailer’s different asset situations after the first sales period, which will determine the retailer needs to order product or make a bank loan or not at the second period.

Case 1:

When the retailer needs to order products with the bank financing at the second period, it implies the following two constraints:

${x}_{1}<{l}_{2}^{b}$ and

${y}_{1}<{w}_{2}{q}_{2}^{b}$. After simplification, we have:

Let

${m}^{b}\left({q}_{1}\right)=\frac{{y}_{0}+{\mathsf{\pi}}_{1}^{b}\left({q}_{1}\right)}{{w}_{2}}+{x}_{0}+{q}_{1}-{l}_{2}^{b}$, from (56) and (57), it can be concluded that the stochastic demand in the first period

${D}_{1}$ needs to meet:

Equation (52) can be written as follows:

In the two-period scenario, Equation (5) shows that the boundary condition satisfies

${\varphi}_{3}\left({x}_{2}\right)=0$ so a retailer’s profit function in the second period of

Case 1 can also be expressed as the retailer’s expected cumulative profit-to-go calculated from the second period as follows:

When

${x}_{1}$ is determined,

${\varphi}_{2}^{b}{\left({x}_{1}\right)}^{\left(1\right)}$ is determined, which can be regarded as a single-period case. When there are two periods with unknown

${x}_{1}$,

${\varphi}_{2}^{b}{\left({x}_{1}\right)}^{\left(1\right)}$ is uncertain. Because of the retailer’s inventory transfer equation as Equation (1), the above formula can be rewritten as follows:

Case 2:

When the retailer needs to order products without bank financing at the second period, it implies the following two constraints similar to Case 1: ${x}_{1}<{l}_{2}^{b}$ and ${y}_{1}>{w}_{2}{q}_{2}^{b}$.

Similar to

Case 1, it can be concluded that

${D}_{1}$ needs to meet Equation (61):

Equation (52) can be written as follows:

Then we get the retailer’s expected cumulative profit-to-go calculated from the second period in

Case 2 as follows:

Case 3:

When the retailer need not order products from the supplier at the second period, it implies

${x}_{1}\ge {l}_{2}^{b}$, i.e.,

${D}_{1}\le {x}_{0}+{q}_{1}-{l}_{2}^{b}$. Under this circumstance, Equation (52) can be written as follows:

Then we get the retailer’s expected cumulative profit-to-go calculated from the second period in

Case 3 as follows:

According to the above three cases which cover all the range regions of the demand in the first period, the retailer’s expected cumulative profit-to-go calculated from the second period can be written as following:

Based on Equation (4), we derive the retailer’s expected cumulative profit-to-go calculated from the first period:

In order to obtain the retailer’s optimal strategy in two-period bank financing situation, we analyze some properties of the retailer’s expected cumulative profit-to-go by Lemma 1.

**Lemma** **1.** The retailer’s expected cumulative profit-to-go ${\varphi}_{1}^{b}\left({x}_{0}\right)$ under two-period bank financing is concave with respect to ${q}_{1}$.

**Proof.** Since

${m}^{b}\left({q}_{1}\right)=\frac{{y}_{0}+{\mathsf{\pi}}_{1}^{b}\left({q}_{1}\right)}{{w}_{2}}+{x}_{0}+{q}_{1}-{l}_{2}^{b}$, we have

$\frac{\partial {m}^{b}\left({q}_{1}\right)}{\partial {q}_{1}}=\frac{{w}_{2}+\frac{\partial {\pi}_{1}^{b}\left({q}_{1}\right)}{\partial {q}_{1}}}{{w}_{2}}$. According to A6, retailer have to order from the supplier and make loans from the bank at the beginning of the first period, which means the retailer’s expected terminal profit at the first period and its first-order differential condition is consistent with the single-period bank credit model in

Section 4.1 as Equations (16) and (17).

According to Equation (67), we have the simplified first-order differential condition of

${\varphi}_{1}^{b}\left({x}_{0}\right)$ as (68):

Then we get the second derivative of the retailer’s expected cumulative profit-to-go calculated from the first period:

Since it is known that ${r}_{f}<{r}_{b}$, we come to the conclusion that the second derivative of the retailer’s expected cumulative profit-to-go calculated from the first period is negative, which means it is concave with respect to ${q}_{1}$. □

According to Lemma 1, the retailer’s expected cumulative profit-to-go calculated from the first period is concave function. According to the nature of concave function, there must be an optimal order quantity ${q}_{1}^{b}$ to maximize the retailer’s expected cumulative profit-to-go. Theorem 6 will give the retailer’s optimal ordering strategy in two-period bank financing.

**Theorem** **6.** Under the two-period bank financing with the assumption that the financial-constrained retailer need to order from the supplier and borrow from the bank at the beginning of the first period, the retailer’s optimal order quantity in the second period is consistent with that in the single period, i.e., ${q}_{2}^{b}=\{\begin{array}{c}{l}_{2}^{b}-{x}_{1},if{x}_{1}{l}_{2}^{b}\\ 0,otherwise\end{array}$ and ${l}_{2}^{b}=\overline{F}{\left(\frac{{w}_{2}\left(1+{r}_{f}\right)}{p}\right)}^{-1}$; the retailer’s optimal order quantity in the first period is ${q}_{1}^{b}={l}_{1}^{b}-{x}_{0}$, in which ${l}_{1}^{b}$ is determined by Equation (70).

**Proof.** According to A6, the financial-constrained retailer need to order from the supplier and borrow from the bank at the beginning of the first period, i.e., ${x}_{0}<{l}_{1}^{b}$ and ${y}_{0}<{w}_{1}{q}_{1}^{b}$.

Since Lemma 1 has proved that the retailer’s expected cumulative profit-to-go

${\varphi}_{1}^{b}\left({x}_{0}\right)$ under two-period bank financing is concave with respect to

${q}_{1}$ with a maximum point. According to Equation (68) in Lemma 1, we derive the retailer’s optimal first order condition by substituting

${x}_{0}+{q}_{1}^{b}={l}_{1}^{b}$ into Equation (68) and obtain Equation (70) after simplification:

In Equation (70), ${m}^{b}\left({l}_{1}^{b}\right)=\frac{{y}_{0}+{\mathsf{\pi}}_{1}^{b}\left({q}_{1}\right)}{{w}_{2}}+{x}_{0}+{q}_{1}-{l}_{2}^{b}=\frac{{y}_{0}+{\mathsf{\pi}}_{1}^{b}\left({l}_{1}^{b}\right)}{{w}_{2}}+{l}_{1}^{b}-{l}_{2}^{b}$ and ${\mathsf{\pi}}_{1}^{b}\left({l}_{1}^{b}\right)=p{l}_{1}^{b}-p{{\displaystyle \int}}_{0}^{{l}_{1}^{b}}F\left({D}_{1}\right)d{D}_{1}+\left(1+{r}_{f}\right)\left({y}_{0}-{w}_{1}{l}_{1}^{b}+{w}_{1}{x}_{0}\right)$.

Due to ${x}_{0}<{l}_{1}^{b}$ implicated by A6, the retailer’s optimal order quantity at the first period satisfies ${q}_{1}^{b}={l}_{1}^{b}-{x}_{0}$, in which ${l}_{1}^{b}$ can be determined by Equation (70). □

After solving the retailer’s problem under two-period bank financing, the supplier game equilibrium two-period bank financing can be drawn as Theorem 7:

**Theorem** **7.** Under the two-period bank financing, the supplier’s optimal wholesale price in the second period is consistent with that in the single period, i.e., ${w}_{2}^{b}=\frac{p{q}_{2}^{b}f\left({l}_{2}^{b}\right)}{\left(1+{r}_{f}\right)}+c$; her optimal wholesale price in the first period is determined by Equations (73) and (74).

**Proof.** Similar to the supplier’s problem under the single-period bank financing analyzed in

Section 4.1, the supplier can use a retailer’s early payment for risk-free investment.

The cumulative profit-to-go calculated from the second period as follows:

From Theorem 2 and Theorem 6, it can be seen that

${l}_{2}^{b}=\overline{F}{\left(\frac{{w}_{2}\left(1+{r}_{f}\right)}{p}\right)}^{-1}$ and

${w}_{2}^{b}=\frac{p{q}_{2}^{b}f\left({l}_{2}^{b}\right)}{\left(1+{r}_{f}\right)}+c$, let

${l}_{1}^{b}\left({w}_{1}\right)={q}_{1}^{b}\left({w}_{1}\right)+{x}_{0}$. The supplier’s expected cumulative profit-to-go calculated from the first period as following:

The optimal first-order condition of supplier’s expected cumulative profit-to-go with respect of

${w}_{1}$ as (73):

It means that the supplier’s optimal wholesale price satisfies Equation (73). According to Equation (68), we get the reaction function of

${l}_{1}^{b}$ and

${w}_{1}$ based on the implicit function theorem:

□

In above, it can be seen that the Stackelberg equilibrium under two-period bank financing is composed of the optimal policy matrix $\left[\begin{array}{cc}{w}_{1}^{b}& {q}_{1}^{b}\\ {w}_{2}^{b}& {q}_{2}^{b}\end{array}\right]$ determined by Theorems 6 and 7.

#### 5.2. Two-Period Supplier Financing Problem

This subsection considers the two-period case under supplier financing to analyze how to maximize the terminal cumulative profits at the end of the second periods with the perspective of the retailer and the supplier, respectively.

Similar to

Section 5.1, we will derive the retailer’s expected cumulative profit-to-go calculated from the first period and obtain its optimal first-order condition as Theorem 8.

**Theorem** **8.** Under the two-period supplier financing with the assumption that the financial-constrained retailer need to order from the supplier and borrow from the supplier at the beginning of the first period, the retailer’s optimal order quantity in the second period is consistent with that in the single period, i.e., ${q}_{2}^{s}=\{\begin{array}{c}{l}_{2}^{s}-{x}_{1},if{x}_{1}{l}_{2}^{s}\\ 0,otherwise\end{array}$ and ${l}_{2}^{s}=\{\begin{array}{c}{\overline{F}}^{-1}\left(\frac{{w}_{2}\left(1+{r}_{s}\right)}{p}\right),{x}_{1}+\frac{{y}_{1}}{{w}_{2}}\in \left(0,{\overline{F}}^{-1}\left(\frac{{w}_{2}\left(1+{r}_{s}\right)}{p}\right)\right)\\ \frac{{y}_{1}}{{w}_{2}}+{x}_{1},{x}_{1}+\frac{{y}_{1}}{{w}_{2}}\in \left({\overline{F}}^{-1}\left(\frac{{w}_{2}\left(1+{r}_{s}\right)}{p}\right),{\overline{F}}^{-1}\left(\frac{{w}_{2}\left(1+{r}_{f}\right)}{p}\right)\right)\\ {\overline{F}}^{-1}\left(\frac{{w}_{2}\left(1+{r}_{f}\right)}{p}\right),{x}_{1}+\frac{{y}_{1}}{{w}_{2}}\in \left({\overline{F}}^{-1}\left(\frac{{w}_{2}\left(1+{r}_{f}\right)}{p}\right),\infty \right)\end{array}$; the retailer’s optimal order quantity in the first period is ${q}_{1}^{s}={l}_{1}^{s}-{x}_{0}$, in which ${l}_{1}^{s}$ is determined by Equation (79).

**Proof.** From Theorem 3, let denote

${l}_{2}^{s}$ represent the retailer’s base-stock level in the second period under supplier financing, we rewrite the above equation as follows:

The retailer’s optimal order quantity satisfies Equation (76):

Since the discussion of retailer’s different financing and operational decisions for his expected cumulative profit-to-go calculated from the second period is similar to

Section 5.1, which also analyzes three cases including ordering and borrowing scenario, ordering and non-borrowing scenario and non-ordering scenario, we omit the discussion procedure for the sake of brevity and derive the retailer’s expected cumulative profit-to-go calculated from the second period as follows:

It is noted that ${m}^{s}\left({q}_{1}\right)=\frac{{y}_{0}+{\mathsf{\pi}}_{1}^{s}\left({q}_{1}\right)}{{w}_{2}}+{x}_{0}+{q}_{1}-{l}_{2}^{s}$, we have $\frac{\partial {m}^{s}\left({q}_{1}\right)}{\partial {q}_{1}}=\frac{{w}_{2}+\frac{\partial {\pi}_{1}^{s}\left({q}_{1}\right)}{\partial {q}_{1}}}{{w}_{2}}$.

Similar to Lemma 1, we come to the conclusion that the second derivative of

${\varphi}_{1}^{s}\left({x}_{0}\right)$ is negative, which means it is concave with respect to

${q}_{1}$. Therefore there must be a

${q}_{1}$ to maximize the retailer’s expected cumulative profit-to-go calculated from the first period

${\varphi}_{1}^{s}\left({x}_{0}\right)$. After some simplification, we obtain the optimal first condition as following:

In (78), ${m}^{s}\left({q}_{1}\right)=\frac{{y}_{0}+{\pi}_{1}^{s}\left({q}_{1}\right)}{{w}_{2}}+{l}_{1}^{s}-{l}_{2}^{s}$ and ${\pi}_{1}^{s}\left({q}_{1}\right)=ph\left({q}_{1}+{x}_{0}\right)-{w}_{1}{q}_{1}-({w}_{1}{q}_{1}-{y}_{0)}{r}_{s}$.

Due to ${x}_{0}<{l}_{1}^{s}$ implicated by A6, the retailer’s optimal order quantity at the first period satisfies ${q}_{1}^{s}={l}_{1}^{s}-{x}_{0}$, in which ${l}_{1}^{s}$ can be determined by Equation (78). □

After solving the retailer’s problem under two-period supplier financing, the supplier game equilibrium of two-period bank financing can be described by Theorem 9:

**Theorem** **9.** Under the two-period supplier financing, the supplier’s optimal wholesale price in the second period is consistent with that in the single period, which can be seen in Theorem 4. Her optimal wholesale price in the first period is determined by Equation (82).

**Proof.** According to Theorem 4, we can know that: when retailer use his partial equity capital with risk-free investment, ${w}_{2}^{s}=\frac{p{q}_{2}^{s}f\left({x}_{1}+{q}_{2}^{s}\right)}{\left(1+{r}_{f}\right)}+c$; when retailer borrows without bankruptcy, ${w}_{2}^{s}=\frac{p{q}_{2}^{s}f\left({x}_{1}+{q}_{2}^{s}\right)}{\left(1+{r}_{s}\right)}+\frac{c}{\left(1+{r}_{s}\right)}$; when retailer borrows with bankruptcy, ${w}_{2}^{s}=\frac{c}{\left(1+{r}_{s}\right)}$.

According to the three cases which cover all the range regions of the demand in the first period which is similar to the retailer problem, the supplier’s expected cumulative profit-to-go calculated from the second period can be written as Equation (79) and the supplier’s total expected cumulative profit-to-go calculated from the first period can be written as Equation (80):

After simplification, we have the first-order differential condition of

${\mathsf{\Phi}}_{1}^{\mathrm{s}}\left({\mathrm{x}}_{0}\right)$ as (81):

Let

$\frac{\partial {\mathsf{\Phi}}_{1}^{s}\left({x}_{0}\right)}{\partial {w}_{1}}=0$ and after simplification, it can be seen that the supplier’s optimal wholesale price satisfies Equation (82):

In (82), $\frac{\partial {\pi}_{1}^{s}\left({q}_{1}\right)}{\partial {q}_{1}}=p\overline{F}\left({l}_{1}^{s}\right)-{w}_{1}\left(1+{r}_{s}\right)\frac{\partial {l}_{1}^{s}}{\partial {w}_{1}}-\left({l}_{1}^{s}-{x}_{0}\right)\left(1+{r}_{s}\right)$.

According to Equation (78), we get the reaction function of

${l}_{1}^{s}$ and

${w}_{1}$ based on the implicit function theorem:

□

In above, it can be seen that the Stackelberg equilibrium under two-period supplier financing is composed of the optimal policy matrix $\left[\begin{array}{cc}{w}_{1}^{s}& {q}_{1}^{s}\\ {w}_{2}^{s}& {q}_{2}^{s}\end{array}\right]$ determined by Theorems 8 and 9.