Reputation compensation for incentive alignment in a supply chain with trade credit under information asymmetry

Abstract

This paper examines a two-period dynamic contracting in a supply chain under information asymmetry, where a supplier sells a product to a retailer via a trade credit contract. It is found that the retailer always prefers to conceal her actual cost information thus signal as a higher-cost type in the first period to pursue a higher information rent, which would decrease the supplier’s profit and thereby the overall profit of the supply chain. To mitigate this ratchet effect, we introduce a reputation compensation mechanism in the two-period trade credit setting. This mechanism could alleviate the information asymmetry to a certain extent as there exists a threshold that incentivizes the retailer to share her true cost information in the earlier period. Moreover, the retailer might claim as a lower-cost type when the supplier offers a relatively higher reputation compensation to take full advantage of her information. Therefore, the supplier should provide trade credit with a reasonable reputation compensation in a two-period setting to enhance his expected profit.

1 Introduction

In turbulent times such as COVID-19 pandemic, small and medium-sized enterprises (SMEs) typically face working capital constraints that could cause supply chain disruptions (Dimson & Sharma, 2021; Rowsell, 2022). Therefore, enterprises are in need of short-term financing to execute their procurement (Eggers, 2020). Trade credit—in which suppliers finance their receivables by providing credit to buying firms—is one of the most frequently adopted short-term financing instruments when enterprises have difficulty accessing loans from bank (Boissay et al., 2020). More than 80% of B2B transactions in the US and the UK are completed with trade credit financing (Seifert et al., 2013). It is found that the proportion of trade credit in the total financing of enterprises is 5.7% in the UK, 8% in the United States, 8.2% in the Germany, 12.5% in the Italy, 15.5% in the France, and 17.9% in the Japan (Chen et al., 2021; Yang & Birge, 2018). It is shown that trade credit in the United States (US) increased by 17.77% in 2018 (Zhi et al., 2022). As an internal financing mechanism in a supply chain, trade credit enables the retailers to extend their payment for the procurement from upstream suppliers, and thereby alleviates the retailers’ working capital shortage. In a decentralized supply chain, the presence of trade credit could motivate the retailer to increase order quantity and improve the supplier’s profit (Bi et al., 2021; Kouvelis & Zhao, 2012).

In a supply chain, suppliers and retailers act as both fully rational economic entities that expect to maximize their own benefits. The realization of optimal profit depends on the interactive decisions-making of the supply chain members. In trade credit, the supplier (he) needs to take into account the reaction of the retailer (she) to his proposed terms. For example, the supplier can influence the retailer’s ordering plan by specifying the credit period of time (Li et al., 2019; Luo & Zhang, 2012; Zhong & Zhou, 2013). So it is non-trivial to determine a trade credit contract that maximizes the supplier’s profit and that is accepted by the retailer. And this matter becomes complicated under asymmetric information in a supply chain (Kerkkamp et al., 2018; Vosooghidizaji et al., 2020).

The majority of research on trade credit assumes that information is public among the supply chain members (Devalkar & Krishnan, 2019; Wang et al., 2021b). However, in practice, suppliers and retailers in supply chains often possess their own private information. For example, retailers rarely share their cost structure with suppliers to receive favorable credit terms and to secure better rights and interests (Luo & Zhang, 2012; Wang et al., 2021b). In the case that the retailer holds private information unobserved by the supplier before signing a contract (i.e. adverse selection), the supplier can improve his situation by designing a menu of contracts for the retailer to choose from (Laffont & Martimort, 2001; Vosooghidizaji et al., 2020). We tackle a supply chain contract design problem under trade credit when the retailer holds her private information about sales cost.

The multi-period cooperation conforms more to the real-world supply chain under trade credit (Chen et al., 2021; Li et al., 2019). In that regard, our model focuses on a dynamic adverse-selection problem under trade credit in a two-period setting. In the first period, the supplier offers a menu of contracts according to his prior belief on the retailer’s cost, and the retailer may choose one of contracts. Once one contract is selected by the retailer, the retailer makes the purchasing and the supplier determines the credit period within which the retailer can delay payment for the purchase. In the second period, the supplier's belief on cost information is updated based on the retailer’s selection in the first period, and the supplier offers a new menu of contracts to the retailer accordingly. We examine a dynamic contracting model with trade credit under information asymmetry to address the following research questions:

• (i) In a two-period trade credit game where a retailer holds private cost information, how could a supplier optimize the dynamic contract to maximize his expected profit?

• (ii) If the retailer take advantage of her private information to pursue a higher information rent, would this affect the value of trade credit and undermine the supplier’s profit?

• (iii) How should the supplier design a reputation compensation mechanism to incentivize the retailer to disclose her private information in an earlier period?

In particular, we consider a supplier who sells his product to a retailer in two periods, and the retailer possesses her private cost information which is cannot be observed by the supplier. In each period, the supplier designs a menu of contracts consisting of his credit period and the retailer’s order quantity, from which the retailer can choose or reject. We construct and analyze the two-period trade credit model with price-dependent demand. Our analysis leads to the following main insights: In the two-period setting, although trade credit can incentivize the retailer to disclose her private information to a certain extent, the retailer always tends to hide the real information in the first period and disguise herself as a cost type higher than her real one in order to acquire greater benefits. If the retailer announces her real cost type in the first period, the dominant supplier could extract supply chain profit in the subsequent contracting process so that the retailer only receives her reservation profit. Consequently, the supplier is unable to make the optimal decisions owing to the information that the retailer conceals. Therefore, a reputation compensation mechanism is proposed and analyzed in the two-period trade credit contract. We find that the reputation compensation can affect the cost type disclosed by the retailer, and there exists a threshold which motivates the retailer to announce her real cost in the first period. If the compensation factor is less than the threshold value, although the retailer still claims to be a certain cost type higher than her real cost, the compensation mechanism does have a positive impact on the retailer’s incentive to share her true cost information, i.e., the magnitude of information asymmetry is decreased. However, when the compensation factor is greater than the threshold value, the retailer would claim as a certain cost type lower than her real cost, as an effort to capture the excess compensation after the information rent is offset.

The remainder of our paper is organized as follows. Section 2 reviews the related literature, followed by the model assumptions and notations in Sect. 3. Section 4 studies the single-period model and Sect. 5 formulates and analyzes the dynamic contracting problem in a two-period setting. We introduce reputation compensation into the two-period model in Sect. 6. Section 7 concludes with a discussion and managerial insights.

2 Literature review

2.1 Trade credit

Our paper is related to the research streams in operations-finance interface (Babich & Kouvelis, 2018; Wang et al., 2021a; Zhao & Huchzermeier, 2015, 2018) focusing on trade credit in supply chain finance (Dekkers et al., 2020; Seifert et al., 2013).

The first research stream primarily optimizes firms’ operations strategies under given trade credit terms in single-firm settings. Haley and Higgins (1973), Goyal (1985), Aggarwal and Jaggi, (1995), Huang (2003), Taleizadeh et al. (2013), Cárdenas-Barrón et al. (2020) and Gupta et al. (2020) develop economic order quantity (EOQ) models to study a retailer’s inventory policy in various settings under trade credit. Chen et al. (2014), Wu et al. (2016) and Tiwari et al. (2022) extend EOQ model for deteriorating items under two-level trade credit. Jaggi et al. (2017) and Feng and Chan (2019) study a firm’s optimal decisions under a price-dependent demand. Roy et al. (2020) present a two-warehouse probabilistic model for deteriorating items with a stochastic demand function under two levels of trade-credit policy. Moreover, one substream of this literature concentrates on supply chain operations and performance under given trade credit terms, which has been investigated from various perspectives, e.g., supply chain coordination (Lee & Rhee, 2011; Phan et al., 2019; Xiao et al., 2017), the comparison between trade credit and other credit contracts (Kouvelis & Zhao, 2012; Cai et al., 2014; Chen, 2015; Chod, 2017; Shi et al., 2021; Zhi et al., 2022), the risk-sharing role of trade credit (Yang & Birge, 2018), the risk attitude of participators (Li et al., 2018; Yan et al., 2019; Yang et al., 2021), the multi-item inventory model for deteriorating items (Pervin et al., 2019), the competition among suppliers (Peura et al., 2017; Ren et al., 2020), and the competition between retailers (Lan et al., 2019; Zhang & Zhang, 2022).

The second research stream examines the interaction between trade credit terms and firms' operational decisions in single-firm settings. For instance, Wu et al. (2014) and Bi et al. (2021) investigate the retailer’s optimal credit period and ordering policy under two-level trade credit and show that granting downstream trade credit can increase sales. Zou and Tian (2020) study the retailer’s optimal ordering and payment fraction strategy under two-level and flexible two-part trade credit policy. Paul et al. (2021) formulate an EOQ model with time dependent deterioration rate and demand depend on price and derive the optimal replenishment time and credit period. Furthermore, Luo and Zhang (2012) and Zhong and Zhou (2013) explore the interaction of the supplier’s trade credit period and the retailer’s lot size in a supply chain. It is shown that the supplier can alter the credit period to entice the retailer to increase order quantity. Kouvelis and Zhao (2012) consider a supply chain with a newsvendor-type retailer and a supplier and find that the supplier has incentive to offer trade credit to the retailer at an interest rate less than or equal to that charged by a bank. Wu et al. (2019), Deng et al. (2021) and Wang et al. (2022) develop supply chain models with a manufacturer and two retailers engaged in Cournot competition and explore the interaction between trade credit and supply chain decisions. Zhang and Chen (2021) examine a supply chain consisting of a risk-averse supplier and a capital-constrained OEM to construct a dyadic closed-loop supply chain and analyze the interactions of the supplier’s wholesale price and the OEM’s prepaid payment proportion and production quantity of new products under the financing portfolio of trade credit and bank loan.

The aforementioned studies assume that full information can be observed by supply chain partners. Our study is closely related to the research stream on trade credit that addresses information asymmetry. Luo and Zhang (2012) derive the optimal trade credit periods when the buyer’s capital cost is asymmetric. Wang et al. (2018) investigate and compare the screening, checking, and insurance mechanisms to address credit default problems when the retailer’s credit level is unobservable. Devalkar and Krishnan (2019) study how trade credit coordinates a supply chain when the buyer cannot observe the supplier’s exerting effort (moral hazard). Wang et al. (2021b) consider the setting where a risk-neutral supplier offers trade credit to a risk-averse retailer, and explore the incentive effect of trade credit on the supply chain’s decisions under information asymmetry. Different from the above research, we explore a supply chain contract design problem in a two-period setting and concentrate on the impact of information asymmetry on the interactions of trade credit terms and operational decisions in dynamic contracting.

Research on supply chain decisions under trade credit in multiple periods setting has been emerging recently. For instance, Li et al. (2019) model the interaction between the supplier's credit terms and the buyer's order quantity in a multi-period setting and implement nonlinear programming to derive the optimal solutions. Chen et al. (2021) focus on the value of trade credit under risk control in a two-period setting, and find that trade credit not only increases the profits of both parties in a supply chain but also reduces the retailer’s default risk as long as the supplier can implement appropriate risk control. In contrast to Li et al. (2019) and Chen et al. (2021), we consider information asymmetry under trade credit contract in a two-period supply chain setting.

2.2 Multi-period contracting under information asymmetry

Moreover, our study is related to the literature on multi-period contracting problems in a supply chain under information asymmetry. Gao and Tian (2018) and Mobini et al. (2019) analyze long-term commitment contracts. Gao and Tian (2018) develop a multi-period incentive model to stimulate the enterprise to exert more effort and to consciously abide by the contract signed with the government in order to ensure that the materials are sufficiently supplied in the event of an emergency. Mobini et al. (2019) design multi-period contracts consisting of a procurement plan plus a side payment to analyze the Economic Lot Sizing (ELS) problem when the retailer possesses private information on customer demand and his cost parameters and a discrete type space for each dimension. In addition, Zhang et al. (2010) study a multi-period supply chain model based on dynamic short-term contracts under asymmetric information on initial inventory and show that the optimal contract can minimizes the retailer’s information advantage given relatively high production and holding costs. Zhang and Zhang (2018) consider a two-period supply chain where the supplier with stochastic cost learning sells products to a retailer who could keep private information on potential market size, and investigate the contract preferences (one-part linear contract and two-part nonlinear contract) for a supply chain. Zhang et al. (2021) consider dynamic contracting in a supply chain where a manufacturer selling a seasonal product to a retailer over two periods by investigating quick response (QR), asymmetric demand information and dynamic contract. It is found that QR capability could hurt the manufacturer under asymmetric information. Miao et al. (2022) focus on the comprehensive influence of commodity deterioration and information asymmetry on the strategic inventory in a two-period supply chain with a dynamic contract. It is shown that a dynamic contract is always better than a commitment contract when the retailer holds the deteriorating strategic inventory under information asymmetry. The key distinctions of our study from the aforementioned literature on dynamic contracting under information asymmetry are as follows: First, we investigate a dynamic contracting problem over two periods under asymmetric information in a trade credit setting. Second, we assume that the retailer holds private information on her cost, whereas Zhang et al. (2010) and Miao et al. (2022) assume that the retailer possesses private information on inventory, and Zhang and Zhang (2018) and Zhang et al (2021) focus on demand information asymmetry. Third, we introduce a reputation compensation mechanism to incentivize the retailer to reveal her actual cost information in the first period. To the best of our knowledge, the exploration on the dynamic contracting under information asymmetry in a trade credit setting is among the earliest efforts in trade credit literature.

In sum, this research contributes to the extant research in the following three aspects. To start with, our paper takes an important first step to examine dynamic contracting of trade credit under information asymmetry. We focus on a two-period adverse selection model and derive the optimal dynamic contract of trade credit. Second, we find that trade credit could not effectively incentivize the retailer to disclose her private cost information in an earlier period. Therefore, the ratchet effect incurs and the retailer captures a higher information rent, while the supplier’s profit decreases and the overall supply chain profitability suffers. Third, we introduce a reputation compensation mechanism to mitigate the information asymmetry in the supply chain, while the retailer could take full advantage of her private information to capture benefits. It is shown that the supplier can provide a trade credit contract with reasonable reputation compensation in a two-period setting to enhance his expected profit.

3 Model formulation

In this paper, we consider a supply chain where one supplier (he) sells a single product to one retailer (she) in the market over two periods. Our modeling assumptions are as follows:

Assumption 1

Both the supplier and retailer are risk-neutral, fully rational economic entities and aim to maximize their expected profits, respectively. In each period, the retailer is allowed to delay payment free of interest, i.e., pay back all the purchase to the supplier within the trade credit period.

Assumption 2

For the sake of model tractability and exploration of managerial insights, we assume demand is price sensitive following the literature on trade credit (Pakhira & Maiti, 2021; Soni, 2013; Thangam & Uthayakumar, 2009; Tsao & Sheen, 2012etc.). The order quantity $q$ can be derived by a linear demand function $q=a-bp$, where $a$ is market size and $b$ is the coefficient of price sensitivity (Market size $a$ is assumed as large enough so that $q$ is non-negative and $a>0,b>0,p<a/b$). The retailer selects either $q$ or $p$, then the other is immediately determined. Without loss of generality, the retailer could place her order according to the demand $q$ and the supplier’s production quantity equals to the retailer’s order quantity $q$ (Corbett & Groote, 2000; Corbett et al., 2004; Zhang & Zhang, 2018).

Assumption 3

The retailer holds private information on her unit sales cost ${C}_{r}$ that cannot be observed by the supplier, yet the distribution function $F\left(\bullet \right)$ of her unit cost with probability density function $f\left(\bullet \right)$, defined on the interval $\left[\underset{\_}{{C}_{r}},\overline{{C}_{r}}\right]$, is known to the supplier. Let $f\left(\bullet \right)/\overline{F}\left(\bullet \right)$ denote the failure rate of ${C}_{r}$, where $\overline{F}\left(\bullet \right)=1-F\left(\bullet \right)$. In order to derive closed-form solutions, we assume that the cost distribution has an increasing failure rate (IFR), i.e., the function $f\left(\bullet \right)/\overline{F}\left(\bullet \right)$ is increasing in ${C}_{r}$, which indicates that the higher the cost ${C}_{r}$, the greater the probability density of failure at cost ${C}_{r}$ (Jin & Zhang, 2021; Lariviere & Porteus, 2001; Wang et al., 2021b).

Assumption 4

We analyze a two-period dynamic contracting of trade credit, i.e., the contract is dynamically constructed at the beginning of each period. In the first period, the supplier designs a menu of contracts given his prior belief distribution of the retailer’s unit sales cost ${C}_{r}$. In the second period, the supplier redesigns another menu of contracts based on the updated belief distribution of ${C}_{r}$ contingent on the retailer choice of the contract in the first period. The exact ${C}_{r}$ remains unchanged in the two-period supply chain transactions (Huo et al., 2008; Zhang & Zhang, 2018).

Assumption 5

For the ease of exploration and focus on the retailer’s key trade-offs in profitability (unit sales cost and wholesale price), we assume that other internal variable cost (e.g., holding cost) of the retailer is zero without loss of generality.

The supplier offers the retailer a trade credit contract, i.e., the supplier allows the retailer to delay payment $wq$ with zero interest rate until the end of credit period $t$ (Feng & Chan, 2019; Jin & Zhang, 2021; Kouvelis & Zhao, 2018). Referring to (Luo, 2007; Luo & Zhang, 2012; Wang et al., 2021b), the supplier’s capital cost of the delayed payment is the opportunity cost, i.e., $wq{i}_{s}t$. While for the retailer, she can utilize the capital to earn $wq{i}_{r}t$. In order to ensure the validity of the solution and the rationality of trade credit, we assume ${i}_{r}>{i}_{s}$. Hence, the retailer’s and the supplier’s profit functions can be written as follows, respectively.

$${\Pi }_{r}=pq+{i}_{r}twq-wq-{C}_{r}q=\frac{1}{b}\big(a-q\big)q+{i}_{r}twq-wq-{C}_{r}q$$

(1)

$${\Pi }_{s}=wq-{i}_{s}twq-{C}_{s}q$$

(2)

Here the retailer's profit ${\Pi }_{r}$ incorporates sales revenue $pq$, return on investment ${i}_{r}twq$, purchase cost $wq$ and sales cost ${C}_{r}q$. The supplier’s profit ${\Pi }_{s}$ consists of income from wholesaling products $wq$, capital cost ${i}_{s}twq$ and production cost ${C}_{s}q$.

4 Single-period model

In the base case, we consider a single-period model (i.e., static contact) following the principal-agent framework in a supply chain, where the supplier acts as the principal and the retailer is the agent. As the supplier cannot observe the retailer’s cost, he designs a menu of contracts $\left\{q,t\left(q\right)\right\}$ to motivate the retailer to reveal her true ${C}_{r}$, specifying a credit period $t$ provided for an order quantity $q$. We parameterize $\left\{q,t\left(q\right)\right\}$ on ${C}_{r},{ C}_{r}\in \left[\underset{\_}{{C}_{r}},\overline{{C}_{r}}\right]$, where offering a $\left\{q({C}_{r}),t\left({C}_{r}\right)\right\}$ menu is equivalent to $\left\{q,t\left(q\right)\right\}$ (Corbett & Groote, 2000; Corbett et al., 2004; Zhang & Zhang, 2018). The supplier can infer ${C}_{r}$ after the retailer chooses a specific ($q({C}_{r}),t\left({C}_{r}\right)$)-pair from the menu.

Based on the principal-agent framework and revelation principle (Corbett et al., 2004; Myerson, 1979; Zhang et al., 2010), there is an optimal menu in which the retailer can choose the contract intended for her information sharing. If the retailer accepts that contract, she would announce her cost ${C}_{r}$, which leads to order quantity $q({C}_{r})$ and credit period $t\left({C}_{r}\right)$.The supplier solves the following problem R to maximize his expected profit.

$$\underset{q,t\left(q\right)}{\mathit{max}}E[{\Pi }_{s}(q)]=E[wq-{i}_{s}t(q)wq-{C}_{s}q$$

(4)

$$\begin{aligned} s.t.\qquad &\quad\;{IC:\Pi }_{r}\left(q\left({C}_{r}\right),t\left({C}_{r}\right)\right)\ge {\Pi }_{r}\left(q\left({\widehat{C}}_{r}\right),t\left({\widehat{C}}_{r}\right)\right), \forall {C}_{r},{\widehat{C}}_{r}\in \left[\underset{\_}{{C}_{r}},\overline{{C}_{r}}\right]\end{aligned}$$

(4)

$$IR: {\Pi }_{r}\left({C}_{r}\right)\ge \underset{\_}{{\Pi }_{r}}$$

(5)

The supplier maximizes his expected profit anticipating the retailer’s response to the contract as expressed in the two constraints. Condition (4) denotes the incentive-compatibility (IC) constraint, which states that the supplier incentivizes the retailer to select the contract designed for her to state her true ${C}_{r}$. Condition (5) is the retailer’s individual-rationality (IR) constraint. The constraint guarantees that the retailer receives at least her reservation profit when she chooses the contract designed to incentivize her information disclosure. For the ease of exploration, we normalize the retailer’s reservation profit as 0.

Proposition 1

When ${i}_{s}<{i}_{r}<2{i}_{s}$, The optimal solutions to Problem R under asymmetric cost information in single period are as follows:

$${q}^{*}\left({C}_{r}\right)=\frac{b}{2}\left(\frac{a}{b}+{i}_{r}{t}^{*}\left({C}_{r}\right)w-w-{C}_{r}\right)$$

(6)

$${t}^{*}\left({C}_{r}\right)=\frac{1}{\left({{2i}_{s}-i}_{r}\right)w}\left[\frac{\mathrm{a}\left({i}_{r}-{i}_{s}\right)}{\mathrm{b}{i}_{r}}+\frac{{i}_{s}}{{i}_{r}}w+\frac{{{i}_{s}-i}_{r}}{{i}_{r}}{C}_{r}-{C}_{s}-\frac{F\left({C}_{r}\right)}{f\left({C}_{r}\right)}\right]$$

(7)

Proof

All proofs are given in the Appendix.

Proposition 1 indicates that the Revelation Principle can effectively motivate the retailer to reveal her real cost information in the single-period supply chain model.

5 Two-period model

5.1 Dynamic contract

In this section, we analyze the case of dynamic contracting in a two-period supply chain. Here ${C}_{ri}$ is the cost information the retailer announces in period $i$. ${\Pi }_{ri}$ and ${\Pi }_{si}$ represent the retailer’s and supplier’s profit in period $i$, respectively. $\left\{{q}_{i}\left(\bullet \right),{t}_{i}\left(\bullet \right)\right\}$ denotes the menu of contracts offered by the supplier in period $i$. The subscripts $i$ refers to period $i=\mathrm{1,2}$.

In dynamic contracting, the supplier offers a menu of contract in each period based on his belief of the retailer’s cost information. For the sake of tractability, we follow the research stream on asymmetric cost information to consider an uniform distribution (Lau et al., 2007; Mukhopadhyay et al., 2008; Yao et al., 2008). Suppose the supplier’s initial belief of the retailer’s sales cost distribution is uniform over $\left[\underset{\_}{{C}_{r}},\overline{{C}_{r}}\right]$ with a p.d.f ${f}_{1}\left({C}_{r}\right)=\frac{1}{\overline{{C}_{r}}-\underset{\_}{{C}_{r}}}$ and a c.d.f ${F}_{1}\left({C}_{r}\right)=\frac{{C}_{r}-\underset{\_}{{C}_{r}}}{\overline{{C}_{r}}-\underset{\_}{{C}_{r}}}$. The supplier designs the trade credit contract $\left\{{q}_{1}\left(\bullet \right),{t}_{1}\left(\bullet \right)\right\}$ correspondingly. After the retailer discloses her cost information ${C}_{r1}$ by selecting the contract $({q}_{1}\left({C}_{r1}\right),{t}_{1}\left({C}_{r1}\right))$ in period 1, supplier updates his belief on the retailer’s cost information in period 2 based on ${C}_{r1}$, and considers that the retailer cost type as uniform distribution $\left[\underset{\_}{{C}_{r}},{C}_{r1}\right]$ (Here it is assumed that the retailer would only signal a higher cost. This case is proved in Sect. 5.2.2.). The distribution function ${F}_{2}\left({C}_{r}\right)=\frac{{C}_{r}-\underset{\_}{{C}_{r}}}{{C}_{r1}-\underset{\_}{{C}_{r}}}$, the density function is ${f}_{2}\left({C}_{r}\right)=\frac{1}{{C}_{r1}-\underset{\_}{{C}_{r}}}$, and the second period trade credit contract $\left\{{{q}_{2}\left(\bullet \right),t}_{2}\left(\bullet \right)\right\}$ is designed. The sequence of events is depicted in Fig. 1.

Fig. 1

Timeline of events

Here, $({q}_{1}\left({C}_{r1}\right),{t}_{1}\left({C}_{r1}\right))$ and $({q}_{2}\left({C}_{r2}\right),{t}_{2}\left({C}_{r2}\right))$ represent the contracts chosen by the retailer in the first period and the second period, respectively. Based on the relevant conclusions in the base case of Sect. 4 and the analyses in Appendix A. The retailer’s profits in Period 1 and Period 2 are as follows, respectively.

$${\Pi }_{r1}={\int }_{{C}_{r1}}^{\overline{{\mathrm{C}}_{\mathrm{r}}}}q\left(\tau \right)\mathrm{d}\tau +\left({C}_{r1}-{C}_{r}\right){q}_{1}\left({C}_{r1}\right)$$

(8)

$${\Pi }_{r2}={\int }_{{C}_{r2}}^{{C}_{r1}}q\left(\tau \right)\mathrm{d}\tau +\left({C}_{r2}-{C}_{r}\right){q}_{2}\left({C}_{r2}\right)$$

(9)

Contingent on the disclosure of retailer’s cost information in different periods, the supplier’s profits in Period 1 and Period 2 are as follows (respectively):

$${\Pi }_{s1}=\frac{1}{b}\left(a-{q}_{1}\left({C}_{r1}\right)\right){q}_{1}\left({C}_{r1}\right)+\left({i}_{r}-{i}_{s}\right){t}_{1}\left({C}_{r1}\right)w{q}_{1}\left({C}_{r1}\right)-{C}_{r1}{q}_{1}\left({C}_{r1}\right)-{C}_{s}{q}_{1}\left({C}_{r1}\right)-{\int }_{{C}_{r1}}^{\overline{{C}_{r}}}q\left(\tau \right)d\tau $$

(10)

$${\Pi }_{s2}=\frac{1}{b}\left(a-{q}_{2}\left({C}_{r2}\right)\right){q}_{2}\left({C}_{r2}\right)+\left({i}_{r}-{i}_{s}\right){t}_{2}\left({C}_{r2}\right)w{q}_{2}\left({C}_{r2}\right)-{C}_{r2}{q}_{2}\left({C}_{r2}\right)-{C}_{s}{q}_{2}\left({C}_{r2}\right)-{\int }_{{C}_{r2}}^{{C}_{r1}}q\left(\tau \right)d\tau $$

(11)

Suppose the discount factor is 1, the retailer’s total profit is $u={\Pi }_{r1}+{\Pi }_{r2}$, and the supplier’s total profit is $\pi ={\Pi }_{s1}+{\Pi }_{s2}$.

5.2 Two-period model

In the two-period setting, the retailer chooses between two strategies of information sharing: (1) disclose her true cost ${C}_{r}$ in the first period; (2) conceal her true cost ${C}_{r}$ in the first period. We would analyze and compare the two-period models in the two cases.

5.2.1 Disclosure of cost information the in period one

If the retailer chooses to reveal her true cost type in the first period, the true cost information can always be revealed to the supplier in the second period (Zhang & Zhang, 2018). Therefore, the decisions of both supply chain members are consistent with the optimal decisions in the base case, i.e.,${C}_{r1}={C}_{r2}={C}_{r}$,${q}_{1}\left({C}_{r1}\right)={q}_{2}\left({C}_{r2}\right)={q}^{*}\left({C}_{r}\right)$, ${t}_{1}\left({C}_{r1}\right)={t}_{2}\left({C}_{r2}\right)={t}^{*}\left({C}_{r}\right)$, and the optimal contracts in both periods are $({q}^{*}\left({C}_{r}\right),{t}^{*}\left({C}_{r}\right))$. The retailer’s respective profits in period 1 and period 2 are ${\Pi }_{r1}={\int }_{{C}_{r}}^{\overline{{\mathrm{C}}_{\mathrm{r}}}}q\left(\tau \right)\mathrm{d}\tau$ and ${\Pi }_{r2}=0$, thus her total profit ${u}_{T}$ is as follows:

$${u}_{T}={\Pi }_{r1}+{\Pi }_{r2}={\int }_{{C}_{r}}^{\overline{{\mathrm{C}}_{\mathrm{r}}}}q\left(\tau \right)\mathrm{d}\tau $$

(12)

The supplier’s profits in period 1 and period 2 are ${\Pi }_{s1}=\frac{1}{b}\left(a-{q}^{*}\left({C}_{r}\right)\right){q}^{*}\left({C}_{r}\right)+\left({i}_{r}-{i}_{s}\right){t}^{*}\left({C}_{r}\right)w{q}^{*}\left({C}_{r}\right)-{C}_{r}{q}^{*}\left({C}_{r}\right)-{C}_{s}{q}^{*}\left({C}_{r}\right)-{\int }_{{C}_{r}}^{\overline{{C}_{r}}}q\left(\tau \right)d\tau$, and ${\Pi }_{s2}=\frac{1}{b}\left(a-{q}^{*}\left({C}_{r}\right)\right){q}^{*}\left({C}_{r}\right)+\left({i}_{r}-{i}_{s}\right){t}^{*}\left({C}_{r}\right)w{q}^{*}\left({C}_{r}\right)-{C}_{r}{q}^{*}\left({C}_{r}\right)-{C}_{s}{q}^{*}\left({C}_{r}\right)$, respectively. The supplier’s total profit ${\pi }_{T}$ is as follows.

$${\pi }_{T}=2\left[\frac{1}{b}\left(a-{q}^{*}\left({C}_{r}\right)\right){q}^{*}\left({C}_{r}\right)+\left({i}_{r}-{i}_{s}\right){t}^{*}\left({C}_{r}\right)w{q}^{*}\left({C}_{r}\right)-{C}_{r}{q}^{*}\left({C}_{r}\right)-{C}_{s}{q}^{*}\left({C}_{r}\right)\right]-{\int }_{{C}_{r}}^{\overline{{C}_{r}}}q\left(\tau \right)d\tau $$

(13)

5.2.2 Concealed cost information the in period 1

Under dynamic contracting in the two periods, if the retailer hides her true cost type in the first period, the supplier redesigns the contracts contingent on updated belief on the retailer’s cost information in the second period. The retailer is reluctant to reveal the true information early on as an effort to capture a higher profit in the two periods (Zhang et al., 2010). We analyze the retailer's information disclosure by maximizing her total profit $u$ in the two periods, i.e. $\underset{{C}_{r1},{C}_{r2}}{\mathit{max}}u=\underset{{C}_{r1},{C}_{r2}}{max}\left({\Pi }_{r1}+{\Pi }_{r2}\right)$.

Proposition 2

The retailer prefers not to reveal her true cost type and thus signals a higher cost type in Period 1, while chooses to share her true cost information in Period 2.

Proof

The proof is given in Appendix B.

In Period 1, there is at least one ${C}_{r1}^{\prime}\in ({C}_{r},\overline{{C}_{r}}]$, i.e., the retailer’s profit when disclosing the cost type ${C}_{r1}^{\prime}$ is greater than the retailer’s profit when disclosing the true cost type ${C}_{r}$. Therefore, the retailer prefers not to reveal her true cost type and signals a higher cost type, which indicates the revelation principle is no longer valid in the static contract (see Sect. 4). In Period 2, the retailer does not have to consider the impact of current decision on subsequent periods, thus only maximizes her profit. Therefore, the retailer could secure a higher return by revealing her true cost information in Period 2.

By analyzing the retailer’s optimal decisions in the two-period signaling game, the following conclusions can be drawn (the parameters in the numerical examples are set as follows: $w=60,q=2000-10p,{i}_{s}=0.1,{i}_{r}=0.15,{C}_{s}=18,{C}_{r}=10,\underset{\_}{{C}_{r}}=8,\overline{{C}_{r}}=48$):

Theorem 1

When the retailer chooses not to signal her true cost information, the disclosed cost must be no smaller than her true cost ${C}_{r}$. The higher the ${C}_{r}$, the higher the cost signaled by the retailer.

When ${C}_{r1}\le {C}_{r}$, i.e., the cost ${C}_{r1}$ disclosed by the retailer in Period 1 falls into the interval $\left[\underset{\_}{{C}_{r}},{C}_{r}\right]$, $\frac{\partial u}{\partial {C}_{r1}}>0$ always holds, i.e. $u$ is increasing in ${C}_{r1}$. In order to maximize her total profit, the retailer chooses to disclose her true cost ${C}_{r}$ and has no incentive to pretend to be a lower cost type. Moreover, we find that there is at least one sub-interval in $({C}_{r},\overline{{C}_{r}}]$ which makes $\frac{\partial u}{\partial {C}_{r1}}>0$. That is, $u$ is increasing in ${C}_{r1}$ in this sub-interval, thus the retailer has the incentive to inflate her cost information. Therefore, if the retailer chooses not to signal her true cost information, the disclosed cost must be no smaller than her true cost ${C}_{r}$, i.e., the lower-cost retailer has the incentive to disguise herself as a higher-cost retailer, while a higher-cost retailer would not pretend to be a lower-cost retailer. This indicates that the supplier no longer considers the case that the retailer’s cost is in the interval $\left[{C}_{r1},\overline{{C}_{r}}\right]$ at the end of Period 1.

As shown in Fig. 2, the higher the true cost, the higher disclosed cost would be. Meanwhile, owing to the upper bound imposed by the retailer’s true cost, the retailer’s cost information disclosed in Period 1 must be within the interval, i.e., ${C}_{r1}^{\prime}=min\left(\frac{100}{3}+\frac{1}{2}{C}_{r},\overline{{C}_{r}}\right)$. In case the retailer’s true cost ${C}_{r}$ is high so that $\left(\frac{100}{3}+\frac{1}{2}{C}_{r}\right)>\overline{{C}_{r}}$, the retailer would disclose her cost as ${C}_{r1}^{\prime}=\overline{{C}_{r}}$.

Fig. 2

The relationship between the retailer’s disclosed cost type in period 1 and her true cost type

Theorem 2

In the two-period game, the retailer tends to disguise true cost in Period 1 while is willing to signal true cost in Period 2, which could lead to the ratchet effect in a supply chain.

Figure 3a and b shows that the retailer achieves profit maximization if she chooses to disclose her true cost (${C}_{r1}={C}_{r}=10$). Since the retailer’s true cost type is disclosed in Period 1, she receives only her reservation profit. If the retailer chooses to signal a higher cost in Period 1 (assuming ${C}_{r1}=13$), her profit decreases by $\Delta {\Pi }_{r1}$ relative to the case of true cost disclosure, yet can increase her profit by $\Delta {\Pi }_{r2}$ in Period 2. $\Delta {\Pi }_{r2}>\Delta {\Pi }_{r1}$ indicates that the retailer can maximize her total profit if she chooses not to reveal her true cost in Period 1, and the profit increment is the retailer’s information rent when her true cost is not disclosed. Therefore, in a two-period collaboration, the retailer tends not to share private information in an earlier period, which leads to the ratchet effect in a supply chain (Wei, 2020).

Fig. 3

a The impact of the retailer’s disclosed cost in period 1 on the retailer’s profit in period 1. b The impact of the retailer’s disclosed cost in period 1 on the retailer’s profit in period 2

Theorem 3

The retailer’s cost is decreasing in the trade credit term. When the retailer does not disclose her true cost type in period 1, she receives a shorter trade credit term.

From Appendix B, $\frac{\partial {t}_{1}\left({C}_{r1}\right)}{\partial {C}_{r1}}<0$ indicates that the trade credit term is decreasing in the retailer’s disclosed cost. The longer the trade credit term provided by the supplier, the more likely the retailer would disclose a lower cost information. In Fig. 4, point A refers to the case when the retailer reveals her true cost ${C}_{r}={C}_{r1}=10$ in Period 1, the trade credit term is ${t}^{*}.$ Point B indicates the retailer with the true cost ${C}_{r}=10$ discloses her cost information ${C}_{r1}=11$ in the first period, the trade credit term is ${t}_{1}^{\mathrm{^{\prime}}}.$ Point C shows that the retailer reveals her true cost ${C}_{r}={C}_{r1}=10$ in Period 1, the trade credit term is ${\mathrm{t}}_{1}^{\prime}$. Since ${t}_{1}^{\prime}>{t}_{1}^{\prime}$ and ${t}^{*}>{t}_{1}^{\prime}$, the trade credit term received by the retailers who disguise cost information is shorter than those who share true cost information. In one word, the incentive mechanism of trade credit has a positive impact on the retailer’s information disclosure. However, combining with Theorem 2, it is shown that the retailer prefers not to share private information in an earlier period to capture a higher information rent. Therefore, trade credit alone cannot incentivize the retailer to disclose her private information in the first period.$t_{1}^{\prime}$ $t_{1}^{^{\prime\prime}}$ $C$ $B$ $A\left( {C_{r1} = C_{r} ,t_{1} = t^{*} } \right)$ In the case where the retailer hides her true cost information in the first period, the disclosed cost, order quantity, and trade credit term in each period are derived as follows (respectively):${C}_{r1}={C}_{r1}^{\prime}>{C}_{r}$,${C}_{r2}={C}_{r}$,${q}_{1}\left({C}_{r1}\right)={q}_{1}\left({C}_{r1}^{\prime}\right)$,${q}_{2}\left({C}_{r2}\right)={q}^{*}\left({C}_{r}\right)$,${t}_{1}\left({C}_{r1}\right)={t}_{1}\left({C}_{r1}^{\prime}\right), {t}_{2}\left({C}_{r2}\right)={t}^{*}\left({C}_{r}\right)$, and the contracts in the two periods are$({q}_{1}\left({C}_{r1}^{\prime}\right),{t}_{1}\left({C}_{r1}^{\prime}\right))$,$({q}^{*}\left({C}_{r}\right),{t}^{*}\left({C}_{r}\right))$, respectively. We derive the retailer’s total profit ${u}_{F}$ and the supplier’s total profit ${\pi }_{F}$ in the case that the retailer hides her cost information in Period 1 and discloses the true cost ${C}_{r}$ in Period 2. Based on Eqs. (8) and (9), the retailer’s total profit ${u}_{F}$ is:

Fig. 4

The relationship between trade credit term and retailer cost type

$${u}_{F}={\int }_{{C}_{r1}^{\prime}}^{\overline{{C}_{r}}}q\left(\tau \right)d\tau +\left({C}_{r1}^{\prime}-{C}_{r}\right){q}_{1}\left({C}_{r1}^{\prime}\right)+{\int }_{{C}_{r}}^{{C}_{r1}^{\prime}}q\left(\tau \right)d\tau ={\int }_{{C}_{r}}^{\overline{{C}_{r}}}q\left(\tau \right)d\tau +\left({C}_{r1}^{\prime}-{C}_{r}\right){q}_{1}\left({C}_{r1}^{\prime}\right)$$

(14)

Based on Eqs. (10) and (11), the supplier’s total profit ${\pi }_{F}$ is:

$$ \begin{aligned}{\pi }_{F} &=\left[\frac{1}{b}\left(a-{q}_{1}\left({C}_{r1}^{\prime}\right)\right){q}_{1}\left({C}_{r1}^{\prime}\right)+\left({i}_{r}-{i}_{s}\right){t}_{1}\left({C}_{r1}^{\prime}\right)w{q}_{1}\left({C}_{r1}^{\prime}\right)-{C}_{r1}^{\prime}{q}_{1}\left({C}_{r1}^{\prime}\right)\right.\\ &\quad \left. -{C}_{s}{q}_{1}\left({C}_{r1}^{\prime}\right)-{\int }_{{C}_{r1}^{\prime}}^{\overline{{C}_{r}}}q\left(\tau \right)d\tau \right]+\left[\frac{1}{b}\left(a-{q}^{*}\left({C}_{r}\right)\right){q}^{*}\left({C}_{r}\right)\right.\\ &\quad\left. + \left({i}_{r}-{i}_{s}\right){t}^{*}\left({C}_{r}\right)w{q}^{*}\left({C}_{r}\right)-{C}_{r}{q}^{*}\left({C}_{r}\right)-{C}_{s}{q}^{*}\left({C}_{r}\right)-{\int }_{{C}_{r}}^{{C}_{r1}^{\prime}}q\left(\tau \right)d\tau \right]\end{aligned} $$

$$=\left[\frac{1}{b}\left(a-{q}_{1}\left({C}_{r1}^{\prime}\right)\right){q}_{1}\left({C}_{r1}^{\prime}\right)+\left({i}_{r}-{i}_{s}\right){t}_{1}\left({C}_{r1}^{\prime}\right)w{q}_{1}\left({C}_{r1}^{\prime}\right)-{C}_{r1}^{\prime}{q}_{1}\left({C}_{r1}^{\prime}\right)-{C}_{s}{q}_{1}\left({C}_{r1}^{\prime}\right)\right]+\left[\frac{1}{b}\left(a-{q}^{*}\left({C}_{r}\right)\right){q}^{*}\left({C}_{r}\right)+\left({i}_{r}-{i}_{s}\right){t}^{*}\left({C}_{r}\right)w{q}^{*}\left({C}_{r}\right)-{C}_{r}{q}^{*}\left({C}_{r}\right)-{C}_{s}{q}^{*}\left({C}_{r}\right)\right]-{\int }_{{C}_{r}}^{\overline{{C}_{r}}}q\left(\tau \right)d\tau $$

(15)

For the retailer, from (12) and (14) it is derived: $\Delta u={u}_{F}-{u}_{T}=\left({C}_{r1}^{\prime}-{C}_{r}\right){q}_{1}\left({C}_{r1}^{\prime}\right)>0$, that is, the retailer can receive a profit increment by not disclosing her true cost. As a result, the base case is no longer efficient. For the supplier, we can derive from (13) and (15): $\Delta \pi ={\pi }_{F}-{\pi }_{T}=\left[\frac{1}{b}\left(a-{q}_{1}\left({C}_{r1}^{\prime}\right)\right){q}_{1}\left({C}_{r1}^{\prime}\right)+\left({i}_{r}-{i}_{s}\right){t}_{1}\left({C}_{r1}^{\prime}\right)w{q}_{1}\left({C}_{r1}^{\prime}\right)-{C}_{r1}^{\prime}q\left({C}_{r1}^{\prime}\right)-{C}_{s}{q}_{1}\left({C}_{r1}^{\prime}\right)\right]-\left[\frac{1}{b}\left(a-{q}^{*}\left({C}_{r}\right)\right){q}^{*}\left({C}_{r}\right)+\left({i}_{r}-{i}_{s}\right){t}^{*}\left({C}_{r}\right)w{q}^{*}\left({C}_{r}\right)-{C}_{r}{q}^{*}\left({C}_{r}\right)-{C}_{s}{q}^{*}\left({C}_{r}\right)\right]<0$. In a two-period setting, the retailer prefers not to share private information in an earlier period, which leads to the ratchet effect and thereby decreases the supplier’s total profit. From the supply chain’s perspective, $\Delta u+\Delta \pi =\left[\frac{1}{b}\left(a-{q}_{1}\left({C}_{r1}^{\prime}\right)\right){q}_{1}\left({C}_{r1}^{\prime}\right)+\left({i}_{r}-{i}_{s}\right){t}_{1}\left({C}_{r1}^{\prime}\right)w{q}_{1}\left({C}_{r1}^{\prime}\right)-{C}_{r}q\left({C}_{r1}^{\prime}\right)-{C}_{s}{q}_{1}\left({C}_{r1}^{\prime}\right)\right]-\left[\frac{1}{b}\left(a-{q}^{*}\left({C}_{r}\right)\right){q}^{*}\left({C}_{r}\right)+\left({i}_{r}-{i}_{s}\right){t}^{*}\left({C}_{r}\right)w{q}^{*}\left({C}_{r}\right)-{C}_{r}{q}^{*}\left({C}_{r}\right)-{C}_{s}{q}^{*}\left({C}_{r}\right)\right]<0$, which indicates that the ratchet effect results in supply chain inefficiency in that the supplier’s profit decrease is larger than the retailer’s profit increment.

Next, we illustrate the impact of ratchet effect on supply chain profitability via numerical examples.

Case 1: The retailer chooses to disclose true cost information in Period 1, i.e., ${C}_{r1}={C}_{r2}={C}_{r}=10$.

Case 2: The retailer chooses not to disclose true cost information in Period 1, and signals her true cost in Period 2, i.e., ${C}_{r1}=12$, ${C}_{r2}={C}_{r}=10$.

Our numerical analysis shows that the retailer’s total profit increases when she does not share true cost information in the first period. For the supplier, the retailer’s moral hazard leads to higher screening costs. Therefore, when the retailer does not signal true cost, the supplier’s profit decreases, which leads to supply chain inefficiency. The retailer has the incentive to disguise true cost information in a two-period game, and thereby leads to the ratchet effect in a supply chain (Table 1).

Table 1 Impact of ratchet effect on supply chain profitability

The root causes to ratchet effect in a supply chain could be summarized as follows: (i) If the retailer chooses to disclose her true cost information at the beginning, then her profit equals to her reservation profit independent of her true cost type. That is, a lower cost supplier could not receive a higher profit than a higher cost one. (ii) In the case of dynamic contract, the dominating supplier would presume the retailer’s cost type of according to her pre-contract selection, redesign the contract to maximize his profit so that the retailer only receives her reservation profit. Therefore, the retailer’s optimal choice is not to reveal her true cost for profit maximization in an earlier period, and thereby leads to the ratchet effect. (iii) If the retailer discloses her true cost in an earlier period, the dominant supplier would utilize this information and incentivize the retailer to lower her selling price. However, the retailer could not lower her sales cost in a short term thus has to order more products, and thereby increases her inventory risk. As a result, the retailer is reluctant to disclose true cost information in an earlier period, and the ratchet effect in a supply chain is also referred to as "whipping the faster cattle".

6 Dynamic contract with reputation compensation

In this section, we consider the case in which the supplier offers a reputation compensation mechanism to the retailer as an incentive to disclose her private information in the first period, i.e., to alleviate the ratchet effect. Provided the long-term relationship of supply chain members, the firms’ effort and reputation would affect their collaboration and thereby supply chain profitability (Abreu & Pearce, 2007; Guo et al., 2011; Liu et al., 2022; Özdoḡan, 2014). The lower the retailer's sales cost, the longer the credit period provided by the supplier, and the more products the retailer orders, the higher profit the supplier would receive. Therefore, the supplier prefers to collaborate with the low-cost retailer. However, the retailer's sales cost is her private information that the supplier cannot observe and is intertemporal related. The retailer transmits her own sales cost information through the contract choice in the first period, which affects the supplier’s belief of the retailer’s sales cost information in the following period, thus the reputation of the retailer is contingent on her sales cost. In general, the lower the sales cost, the higher her reputation, vice versa. Hence, the supplier would provide a reputation compensation according to sales cost level of the retailer to mitigate the impact of ratchet effect.

We denote the reputation compensation as $r\left({\widetilde{C}}_{r1}\right)=k{\int }_{{\widetilde{C}}_{r1}}^{\overline{{C}_{r}}}q\left(\tau \right)d\tau , (\frac{\partial r\left({\widetilde{C}}_{r1}\right)}{\partial {\widetilde{C}}_{r1}}=- kq\left({\widetilde{C}}_{r1}\right)<0$), where $k\ge 0$ is the compensation ratio. ${\widetilde{C}}_{r1}$,${\widetilde{C}}_{r2}$ denote the cost information announced by the retailer in Period 1 and Period 2, respectively. Next, we analyze the impact of $k$ on the retailer’s information disclosure.

In the presence of reputation compensation, upon the analysis in base case and Appendix, the supplier and the retailer’s profits are as follows, respectively:

$$\Pi =\left[\frac{1}{b}\left(a-{q}_{1}\left({\widetilde{C}}_{r1}\right)\right){q}_{1}\left({\widetilde{C}}_{r1}\right)+\left({i}_{r}-{i}_{s}\right){t}_{1}\left({\widetilde{C}}_{r1}\right)w{q}_{1}\left({\widetilde{C}}_{r1}\right)-{\widetilde{C}}_{r1}{q}_{1}\left({\widetilde{C}}_{r1}\right)-{C}_{s}{q}_{1}\left({\widetilde{C}}_{r1}\right)--k{\int }_{{\widetilde{C}}_{r1}}^{\overline{{C}_{r}}}q\left(\tau \right)d\tau \right]+\left[\frac{1}{b}\left(a-{q}_{2}\left({\widetilde{C}}_{r2}\right)\right){q}_{2}\left({\widetilde{C}}_{r2}\right)+\left({i}_{r}-{i}_{s}\right){t}_{2}\left({\widetilde{C}}_{r2}\right)w{q}_{2}\left({\widetilde{C}}_{r2}\right)-{\widetilde{C}}_{r2}{q}_{2}\left({\widetilde{C}}_{r2}\right)-{C}_{s}{q}_{2}\left({\widetilde{C}}_{r2}\right)-{\int }_{{\widetilde{C}}_{r2}}^{\overline{{C}_{r}}}q\left(\tau \right)d\tau \right]$$

(16)

$$U=k{\int }_{{\widetilde{C}}_{r1}}^{\overline{{C}_{r}}}q\left(\tau \right)d\tau +\left({\widetilde{C}}_{r1}-{C}_{r}\right){q}_{1}\left({\widetilde{C}}_{r1}\right)+{\int }_{{\widetilde{C}}_{r2}}^{\overline{{\mathrm{C}}_{\mathrm{r}}}}q\left(\tau \right)\mathrm{d}\tau +\left({\widetilde{C}}_{r2}-{C}_{r}\right){q}_{2}\left({\widetilde{C}}_{r2}\right)$$

(17)

Based on a principal-agent framework (Corbett & Groote, 2000; Corbett et al., 2004; Myerson, 1979), the supplier’s optimization problem under reputation compensation (P1) is:

$$ \begin{aligned}\underset{{\widetilde{C}}_{r1},{\widetilde{C}}_{r2},k}{\mathit{max}}\Pi & =\underset{{\widetilde{C}}_{r1},{\widetilde{C}}_{r2},k}{\mathit{max}}\Bigg\{\frac{1}{b}\left(a-{q}_{1}\left({\widetilde{C}}_{r1}\right)\right){q}_{1}\left({\widetilde{C}}_{r1}\right)+\left({i}_{r}-{i}_{s}\right){t}_{1}\left({\widetilde{C}}_{r1}\right)w{q}_{1}\left({\widetilde{C}}_{r1}\right)-{\widetilde{C}}_{r1}{q}_{1}\left({\widetilde{C}}_{r1}\right)\\ &\quad -{C}_{s}{q}_{1}\left({\widetilde{C}}_{r1}\right)+\frac{1}{b}\left(a-{q}_{2}\left({\widetilde{C}}_{r2}\right)\right){q}_{2}\left({\widetilde{C}}_{r2}\right)+\left({i}_{r}-{i}_{s}\right){t}_{2}\left({\widetilde{C}}_{r2}\right)w{q}_{2}\left({\widetilde{C}}_{r2}\right)\\ &\quad -{\widetilde{C}}_{r2}{q}_{2}\left({\widetilde{C}}_{r2}\right)-{C}_{s}{q}_{2}\left({\widetilde{C}}_{r2}\right)-{\int }_{{\widetilde{C}}_{r2}}^{\overline{{C}_{r}}}q\left(\tau \right)d\tau -k{\int }_{{\widetilde{C}}_{r1}}^{\overline{{C}_{r}}}q\left(\tau \right)d\tau \Bigg\}\end{aligned}$$

(18)

s.t.

$$IC: \left({\widetilde{C}}_{r1}^{*},{\widetilde{C}}_{r2}^{*}\right)\in \mathit{arg}maxU=\mathit{arg}max\left\{{\int }_{{\widetilde{C}}_{r2}}^{\overline{{C}_{r}}}q\left(\tau \right)d\tau +\left({\widetilde{C}}_{r1}-{C}_{r}\right){q}_{1}\left({\widetilde{C}}_{r1}\right)+\left({\widetilde{C}}_{r2}-{C}_{r}\right){q}_{2}\left({\widetilde{C}}_{r2}\right)+ k{\int }_{{\widetilde{C}}_{r1}}^{\overline{{C}_{r}}}q\left(\tau \right)d\tau \right\}, \forall {C}_{r},{\widetilde{C}}_{r1},{\widetilde{C}}_{r1}\in \left[\underset{\_}{{C}_{r}},\overline{{C}_{r}}\right]$$

(19)

$$IR: U\ge {\int }_{{C}_{r}}^{\overline{{\mathrm{C}}_{\mathrm{r}}}}q\left(\tau \right)\mathrm{d}\tau $$

(20)

Equation (18) denotes that the supplier maximizes his total expected profit in the two periods contingent on the costs ${\widetilde{C}}_{r1},{\widetilde{C}}_{r2}$ indicated by the retailer in each period and $k$ offered by himself. Equation (19) refers to the retailer’s incentive-compatibility (IC) constraint: she would choose ${\widetilde{C}}_{r1}^{*}$ and ${\widetilde{C}}_{r2}^{*}$ to maximize her expected profit. Inequality (20) represents the retailer’s individual rationality (IR) constraint: her profit must be no smaller than ${\int }_{{C}_{r}}^{\overline{{\mathrm{C}}_{\mathrm{r}}}}q\left(\tau \right)\mathrm{d}\tau$, i.e., her profit without reputation compensation.

Our previous analysis shows that the retailer’s profit is no smaller than ${\int }_{{C}_{r}}^{\overline{{\mathrm{C}}_{\mathrm{r}}}}q\left(\tau \right)\mathrm{d}\tau$, thus IR always holds. The first-order condition of (17) is: $\frac{\partial U}{\partial {\widetilde{C}}_{r2}}=\left({\widetilde{C}}_{r2}-{C}_{r}\right)\frac{\partial {q}_{2}\left({\widetilde{C}}_{r2}\right)}{\partial {\widetilde{C}}_{r2}}=0$. Since $\frac{\partial {q}_{2}\left({\widetilde{C}}_{r2}\right)}{\partial {\widetilde{C}}_{r2}}<0$,${\widetilde{C}}_{r2}^{*}={C}_{r}$, the retailer does not have to consider the impact of information disclosure in the second period, thus would share her true cost information to maximize her profit. The retailer’s contract choice is $\left({q}_{2}^{*}\left({\widetilde{C}}_{r2}^{*}\right),{t}_{2}^{*}\left({\widetilde{C}}_{r2}^{*}\right)\right)$, here ${q}_{2}^{*}\left({\widetilde{C}}_{r2}^{*}\right)={q}_{2}^{*}\left({C}_{r}\right)=\frac{b}{2}\left(\frac{a}{b}+{i}_{r}{t}_{2}^{*}\left({C}_{r}\right)w-w-{C}_{r}\right)$, and ${t}_{2}^{*}\left({\widetilde{C}}_{r2}^{*}\right)={t}_{2}^{*}\left({C}_{r}\right)=\frac{1}{\left({{2i}_{s}-i}_{r}\right)w}\left[\frac{\mathrm{a}\left({i}_{r}-{i}_{s}\right)}{\mathrm{b}{i}_{r}}+\frac{{i}_{s}}{{i}_{r}}w+\frac{{{i}_{s}-i}_{r}}{{i}_{r}}{C}_{r}-{C}_{s}-\frac{F\left({C}_{r}\right)}{f\left({C}_{r}\right)}\right]$.

Since ${\widetilde{C}}_{r2}={C}_{r}$, (P1) can be reformulated as (P2):

$$\underset{{\widetilde{C}}_{r1},k}{max}\Pi =\underset{{\widetilde{C}}_{r1},k}{\mathit{max}}\left\{\frac{1}{b}\left(a-{q}_{1}\left({\widetilde{C}}_{r1}\right)\right){q}_{1}\left({\widetilde{C}}_{r1}\right)+\left({i}_{r}-{i}_{s}\right){t}_{1}\left({\widetilde{C}}_{r1}\right)w{q}_{1}\left({\widetilde{C}}_{r1}\right)-{\widetilde{C}}_{r1}{q}_{1}\left({\widetilde{C}}_{r1}\right)-{C}_{s}{q}_{1}\left({\widetilde{C}}_{r1}\right)-k{\int }_{{\widetilde{C}}_{r1}}^{\overline{{C}_{r}}}q\left(\tau \right)d\tau \right\}$$

(21)

s.t.

$$ \begin{aligned}{\widetilde{C}}_{r1}^{*}\in \mathit{arg}maxU &=\mathit{arg}max\Big\{{\int }_{{C}_{r}}^{\overline{{C}_{r}}}q\left(\tau \right)d\tau +\left({\widetilde{C}}_{r1}-{C}_{r}\right){q}_{1}\left({\widetilde{C}}_{r1}\right)\\ &\quad +k{\int }_{{\widetilde{C}}_{r1}}^{\overline{{C}_{r}}}q\left(\tau \right)d\tau \Big\} \end{aligned} $$

(22)

The second order partial derivative of $U$ with regard to (w.r.t.) ${\widetilde{C}}_{r1}$ is: $\frac{{\partial }^{2}U}{\partial {\widetilde{C}}_{r1}^{2}}=\left(2-k\right)\frac{\partial {q}_{1}\left({\widetilde{C}}_{r1}\right)}{\partial {\widetilde{C}}_{r1}}+\left({\widetilde{C}}_{r1}-{C}_{r}\right)\frac{{\partial }^{2}{q}_{1}\left({\widetilde{C}}_{r1}\right)}{\partial {\widetilde{C}}_{r1}^{2}}$. Since $\frac{\partial {q}_{1}\left({\widetilde{C}}_{r1}\right)}{\partial {\widetilde{C}}_{r1}}=\frac{-b{i}_{r}}{2\left({{2i}_{s}-i}_{r}\right)}\bullet d\left(\frac{F\left({\widetilde{C}}_{r1}\right)}{f\left({\widetilde{C}}_{r1}\right)}\right),{\widetilde{C}}_{r1}$ is evenly distributed within $\left[\underset{\_}{{C}_{r}},\overline{{C}_{r}}\right]$, thus $d\left(\frac{F\left({\widetilde{C}}_{r1}\right)}{f\left({\widetilde{C}}_{r1}\right)}\right)=1$, so $\frac{{\partial }^{2}{q}_{1}\left({\widetilde{C}}_{r1}\right)}{\partial {\widetilde{C}}_{r1}^{2}}=0$, $\frac{{\partial }^{2}U}{\partial {\widetilde{C}}_{r1}^{2}}=\left(2-k\right)\frac{\partial {q}_{1}\left({\widetilde{C}}_{r1}\right)}{\partial {\widetilde{C}}_{r1}}$. If there exists the optimal ${\widetilde{C}}_{r1}$ that maximizes the retailer’s total profit, $\frac{{\partial }^{2}U}{\partial {\widetilde{C}}_{r1}^{2}}<0$, i.e., $k<2$. Therefore, if $k<2$, the first order condition (FOC) of $U$ w.r.t. ${\widetilde{C}}_{r1}$ indicates that the retailer’s optimal disclosed cost is ${\widetilde{C}}_{r1}^{*}\left(k\right)=\frac{1-\mathrm{k}}{2-k}\left[\frac{a{i}_{s}}{b{i}_{r}}+\frac{{i}_{s}-{i}_{r}}{{i}_{r}}\left(w+{C}_{r}\right)+\underset{\_}{{C}_{r}}-{C}_{s}\right]+\frac{k}{2-k}{C}_{r}$, thus her optimal contract choice is $\left({q}_{1}^{*}\left({\widetilde{C}}_{r1}^{*}\left(k\right)\right),{t}_{1}^{*}\left({\widetilde{C}}_{r1}^{*}\left(k\right)\right)\right)$, where ${q}_{1}^{*}\left({\widetilde{C}}_{r1}^{*}\left(k\right)\right)=\frac{b}{2}\left(\frac{a}{b}+{i}_{r}{t}_{1}^{*}\left({\widetilde{C}}_{r1}^{*}\left(k\right)\right)w-w-{C}_{r}\right)$, and ${t}_{1}^{*}\left({\widetilde{C}}_{r1}^{*}\left(k\right)\right)=\frac{1}{\left({{2i}_{s}-i}_{r}\right)w}\Bigg[\frac{\mathrm{a}\left({i}_{r}-{i}_{s}\right)}{\mathrm{b}{i}_{r}}+\frac{{i}_{s}}{{i}_{r}}w+\frac{{{i}_{s}-i}_{r}}{{i}_{r}}{C}_{r}-{C}_{s}+\underset{\_}{{C}_{r}}-\frac{1-k}{2-k}\left(\frac{a{i}_{s}}{b{i}_{r}}+\frac{{i}_{s}-{i}_{r}}{{i}_{r}}\left(w+{C}_{r}\right)+\underset{\_}{{C}_{r}}-{C}_{s}\right)-\frac{k}{2-k}{C}_{r}\Bigg]$.

Since $\frac{\partial U}{\partial {\widetilde{C}}_{r1}}=\left(1-k\right){q}_{1}\left({\widetilde{C}}_{r1}\right)+\left({\widetilde{C}}_{r1}-{C}_{r}\right)\frac{\partial {q}_{1}\left({\widetilde{C}}_{r1}\right)}{\partial {\widetilde{C}}_{r1}}$, the compensation factor $k$ affects the retailer’s revealed cost type as follows:

• (1) When $k=0$, i.e., compensation is zero, ${\widetilde{C}}_{r1}^{*}={C}_{r1}^{\prime}>{C}_{r}$, the retailer tends to disclose information as a higher cost type retailer (Sect. 5.2);

• (2) When $0<k<1$, if ${\widetilde{C}}_{r1}\le {C}_{r}$, then $\frac{\partial U}{\partial {\widetilde{C}}_{r1}}$ is strictly larger than one, i.e., the retailer’s profit is strictly increasing in ${\widetilde{C}}_{r1}$ within the interval $\left[\underset{\_}{{C}_{r}},{C}_{r}\right]$. Therefore, there exists at least one ${\widetilde{C}}_{r1}^{*}\in ({C}_{r}, \overline{{C}_{r}}]$ which makes the reatiler’s profit when disclosing cost ${\widetilde{C}}_{r1}^{*}$ is higher than that when signaling ${C}_{r}$. That is, when $0<k<1$, ${\widetilde{C}}_{r1}^{*}>{C}_{r}$ exists, thus the retailer has the incentive to signal a higher cost than her true one.

• (3) When $k=1$, $\frac{\partial U}{\partial {\widetilde{C}}_{r1}}=\left({\widetilde{C}}_{r1}-{C}_{r}\right)\frac{\partial {q}_{1}\left({\widetilde{C}}_{r1}\right)}{\partial {\widetilde{C}}_{r1}}$. If ${\widetilde{C}}_{r1}^{*}={C}_{r}$, the retailer maximizes her total profit in two periods. When the compensation ratio provided by the supplier is 1, the retailer would choose to reveal the true cost information in Period 1, and thereby eliminate the ratchet effect in the supply chain.

• (4) When $1<k<2$,$\frac{\partial U}{\partial {\widetilde{C}}_{r1}}=\left(1-k\right){q}_{1}\left({\widetilde{C}}_{r1}\right)+\left({\widetilde{C}}_{r1}-{C}_{r}\right)\frac{\partial {q}_{1}\left({\widetilde{C}}_{r1}\right)}{\partial {\widetilde{C}}_{r1}}$. If ${\widetilde{C}}_{r1}\ge {C}_{r}$, then $\frac{\partial U}{\partial {\widetilde{C}}_{r1}}$ is strictly less than zero, thus there is at least one ${\widetilde{C}}_{r1}^{*}$ (and ${\widetilde{C}}_{r1}^{*}<{C}_{r}$) that maximizes the retailer’s total profit. Hence, the retailer would choose to signal a cost type lower than her true cost for profit maximization, in response to the higher reputation compensation provided by the supplier.

We examine the reputation compensation to incentivize the retailer, and derive the follwing parameters: ${C}_{r1}^{\prime}=\mathrm{min}\left(\frac{100}{3}+\frac{1}{2}{C}_{r},\overline{{C}_{r}}\right), {\widetilde{C}}_{r1}^{*}\left(k\right)=\frac{4k-1}{6-3k}{C}_{r}+\frac{70\left(1-k\right)}{2-k}$, and $\forall {C}_{r1}^{\prime}$, ${\widetilde{C}}_{r1}^{*}\in \left[\underset{\_}{{C}_{r}},\overline{{C}_{r}}\right]$; Here ${C}_{r1}^{\prime}$ and ${\widetilde{C}}_{r1}^{*}$ are (respectively) the disclosed costs before and after the retailer amends the contract when unit product sales cost is ${C}_{r}$.

Figure 5 shows that when $k=1$ and ${\widetilde{C}}_{r1}^{*}={C}_{r}$, the retailer chooses to reveal her true cost type in Period 1. In addition, ${\widetilde{C}}_{r1}^{*}\le {C}_{r1}^{\prime}$ always holds within the interval of compensation ratio $k$, which indicates that the supplier’s reputation compensation to the retailer has a positive impact on the her motivation to share true cost information. In the presence of reputation compensation, regardless of its magnitude, the retailer prefers to disclose a lower cost type. Therefore, reputation compensation could alleviate the adverse impact of ratchet effect in a supply chain.

Fig. 5

The relationship between ${\widetilde{C}}_{r1}^{*}$ and compensation ratio $k$

When the compensation factor provided by the supplier is small thus the incentive is not strong enough, the retailer would disclose a cost type higher than her true one to receive additional information rent, the ratchet effect still exists. When the compensation ratio offered by the supplier is large enough to compensate the retailer’s information rent, the retailer would choose to signal a lower cost type in order to secure the reputation compensation. Only if the compensation factor is 1, the retailer would choose to reveal her true cost information, and thereby eliminates the ratchet effect in the supply chain.

7 Conclusions

This research takes an important first step to examine dynamic contracting of trade credit under information asymmetry. We study a two-period game with dynamic contract of trade credit between a supplier and a retailer under asymmetric information on the latter’s sales cost. We find that in the two-period dynamic contracting, the derived contract is suboptimal as the retailer always prefers not to share her true cost information in the first period while discloses her true cost in second period. Therefore, the ratchet effect incurs and the retailer captures a higher information rent, while the supplier’s profit decreases and the overall supply chain profitability suffers. To mitigate the ratchet effect, we introduce a reputation compensation mechanism in the trade credit contract, and illustrate that this mechanism can alleviate the ratchet effect in a supply chain as there exists a certain threshold that incentives the retailer to reveal her true cost information in an earlier period. However, when the reputation compensation is higher than a threshold, the retailer might claim that to be a cost type lower than her real cost. Hence, the supplier should offer a dynamic trade credit contract with reasonable reputation compensation in a two-period setting under asymmetric information to enhance his expected profit.

The research could be extended in the following aspects. First, for the sake of tractability, we employ the uniform distribution of sales cost following the extant literature on asymmetric cost information. Therefore, other types of random cost distribution can also be considered under asymmetric information as an extension of our model. Moreover, in light of the significant model complexity, numerical analysis is adopted in certain solutions for approximation. Hence, closed-form solutions could be explored in future research. Third, the market demand can be uncertain thus fluctuating in reality, so it could be interesting to investigate into a two-period dynamic model in the trade credit setting where a newsvendor-like retailer faces a stochastic demand, which could enrich the operations-finance interface literature on trade credit. Fourth, a more complex supply chain structure can be explored in the presence of trade credit under information asymmetry. Finally, a more general dynamic contracting model in trade credit can be considered by extending this research in a multi-period setting.

Appendices

Appendix A: Proof of Proposition 1

We solve the model R using Karush–Kuhn–Tucker (KKT) conditions (Wang et al., 2021b).

If the retailer chooses to disclose her true cost in order to maximize her profit, ${\Pi }_{r}\left({C}_{r}\right)=\underset{{\widehat{C}}_{r}}{\mathit{max}}{\Pi }_{r}\left({\widehat{C}}_{r},{C}_{r}\right)$ is the optimal profit of a ${C}_{r}$-type retailer, here the retailer’s disclosed cost ${\widehat{C}}_{r}$ is a function of her true cost type ${C}_{r}$, i.e.,${\widehat{C}}_{r}={\widehat{C}}_{r}\left({C}_{r}\right),{\Pi }_{r}\left({\widehat{C}}_{r},{C}_{r}\right)=\frac{1}{b}\left(a-q\left({\widehat{C}}_{r}\right)\right)q\left({\widehat{C}}_{r}\right)+{i}_{r}t\left({\widehat{C}}_{r}\right)wq\left({\widehat{C}}_{r}\right)-wq\left({\widehat{C}}_{r}\right)-{C}_{r}q\left({\widehat{C}}_{r}\right)$, using the revelation principle, the FOC of ${\Pi }_{r}\left({\widehat{C}}_{r},{C}_{r}\right)$ w.r.t. ${\widehat{C}}_{r}$ is zero, by envelope theorem of maximization, when the retailer discloses.

$${\widehat{C}}_{r}={C}_{r}, \frac{\mathrm{d}{\Pi }_{r}\left({C}_{r}\right)}{\mathrm{d}{C}_{r}}=-q\left({C}_{r}\right)$$

(A1)

The second order condition of the retailer’s profit gives $\frac{{\partial }^{2}{\Pi }_{r}}{\partial {q}^{2}}=-\frac{2}{b}<0$, which shows that the retailer’s profit ${\Pi }_{r}$ is a concave function of her order quantity $q$, the FOC gives $\frac{\partial {\Pi }_{r}}{\partial q}=\frac{a}{b}-\frac{2}{b}q+{i}_{r}tw-w-{C}_{r}=0$, thus ${q}^{*}=\frac{b}{2}\left(\frac{a}{b}+{i}_{r}tw-w-{C}_{r}\right)$.

Since $\forall {\widehat{C}}_{r}\in \left[\underset{\_}{{C}_{r}},\overline{{C}_{r}}\right]$ and ${\Pi }_{r}\left({C}_{r}\right)\ge 0$, ${\Pi }_{r}\left(\overline{{C}_{r}}\right)\ge 0$ holds. The integral of (A1) shows that: ${\Pi }_{r}\left({C}_{r}\right)={\Pi }_{r}\left(\overline{{\mathrm{C}}_{\mathrm{r}}}\right)+{\int }_{\overline{{\mathrm{C}}_{\mathrm{r}}}}^{{C}_{r}}\left[-q\left(\tau \right)\right]\mathrm{d}\tau +{\int }_{{C}_{r}}^{\overline{{\mathrm{C}}_{\mathrm{r}}}}q\left(\tau \right)\mathrm{d}\tau ={\int }_{{C}_{r}}^{\overline{{\mathrm{C}}_{\mathrm{r}}}}q\left(\tau \right)\mathrm{d}\tau$. Hence (3) can be transformed as ${\Pi }_{r}\left({C}_{r}\right)\ge {\int }_{{C}_{r}}^{\overline{{\mathrm{C}}_{\mathrm{r}}}}q\left(\tau \right)\mathrm{d}\tau$. Based on the Eqs. (1) and (2) in Sect. 2, according to ${\Pi }_{r}=pq+{i}_{r}twq-wq-{C}_{r}q=\frac{1}{b}\left(a-q\right)q+{i}_{r}twq-wq-{C}_{r}q$, we can derive that $wq=\frac{1}{b}\left(a-q\right)q+{i}_{r}twq-{C}_{r}q-{\Pi }_{r}$, then ${\Pi }_{s}=wq-{i}_{s}twq-{C}_{s}q$=$\frac{1}{b}\left(a-q\right)q+{i}_{r}twq-{C}_{r}q-{\Pi }_{r}-{i}_{s}twq-{C}_{s}q=\frac{1}{b}\left(a-q\right)q+({i}_{r}-{i}_{s})twq-{C}_{r}q-{\Pi }_{r}-{C}_{s}q$.

The optimization problem R in the base case can be rewritten as:

$$ \begin{aligned}\underset{t\left(\bullet \right)}{\mathit{max}}E{\Pi }_{s} & =\underset{t\left(\bullet \right)}{\mathit{max}}{\int }_{\underset{\_}{{C}_{r}}}^{\overline{{C}_{r}}}\Bigg[\frac{1}{b}\left(a-q\left({C}_{r}\right)\right)q\left({C}_{r}\right)+\left({i}_{r}-{i}_{s}\right)t\left({C}_{r}\right)wq\left({C}_{r}\right)-{C}_{r}q\left({C}_{r}\right)\\ &\quad -{\Pi }_{r}\left({C}_{r}\right)- {C}_{s}q\left({C}_{r}\right)\Bigg]f\left({C}_{r}\right)d{C}_{r}\end{aligned} $$

s.t.

$$IC: {\Pi }_{r}\left(q\left({C}_{r}\right),t\left({C}_{r}\right)\right)\ge {\Pi }_{r}\left(q\left({\widehat{C}}_{r}\right),t\left({\widehat{C}}_{r}\right)\right), \forall {C}_{r},{\widehat{C}}_{r}\in \left[\underset{\_}{{C}_{r}},\overline{{C}_{r}}\right]$$

$$IR:{\Pi }_{r}\left({C}_{r}\right)\ge {\int }_{{C}_{r}}^{\overline{{\mathrm{C}}_{\mathrm{r}}}}q\left(\tau \right)\mathrm{d}\tau $$

We establish the Lagrangian function as:

$$ \begin{aligned}L&={\int }_{\underset{\_}{{C}_{r}}}^{\overline{{C}_{r}}}\Bigg[\frac{1}{b}\left(a-q\left({C}_{r}\right)\right)q\left({C}_{r}\right)+\left({i}_{r}-{i}_{s}\right)t\left({C}_{r}\right)wq\left({C}_{r}\right)-{C}_{r}q\left({C}_{r}\right)\\ &\quad -{\Pi }_{r}\left({C}_{r}\right)-{C}_{s}q\left({C}_{r}\right)\Bigg]f\left({C}_{r}\right)d{C}_{r}+\lambda \big[{\Pi }_{r}\left(q\left({C}_{r}\right),t\left({C}_{r}\right)\right)\\ &\quad -{\Pi }_{r}\left(q\left({\widehat{C}}_{r}\right),t\left({\widehat{C}}_{r}\right)\right)\big]+\mu \left[{\Pi }_{r}\left({C}_{r}\right)-{\int }_{{C}_{r}}^{\overline{{C}_{r}}}q\left(\tau \right)d\tau \right] \end{aligned} $$

(A2)

Here, $\lambda$ and $\mu$ be Lagrangian multipliers corresponding to IC constraints and IR constraints respectively. The FOC of (A-2) w.r.t. $t\left({C}_{r}\right)$ shows:$\frac{\partial L}{\partial t\left({C}_{r}\right)}=-{i}_{s}wq\left({C}_{r}\right)+\lambda \left[{i}_{r}wq\left({C}_{r}\right)-{i}_{r}wq\left({\widehat{C}}_{r}\right)\right]+\mu {i}_{r}wq\left({C}_{r}\right)=0$, using Kuhn-Tucker conditions, it is derived $\lambda =0$ (If $\lambda >0$, according to the Kuhn-Tucker conditions, the corresponding inequality constraints would become equations, which indicates that the retailer’s profit when not disclosing true cost information is equivalent to that when disclosing true cost information. This makes it impossible for the supplier to distinguish the cost type of the retailer and provide corresponding incentive, and thereby affects supply chain efficiency. Therefore, $\lambda >0$ does not hold, so $\lambda =0$) Since $\lambda =0$, $\frac{\partial L}{\partial t\left({C}_{r}\right)}=-{i}_{s}wq\left({C}_{r}\right)+\mu {i}_{r}wq\left({C}_{r}\right)=0$, thus $\mu =\frac{{i}_{s}}{{i}_{r}}$. As $0<{i}_{s}<{i}_{r}<1$, thus $\mu >0$, That is, the IR constraint holds in the transformed optimization problem, i.e., ${\Pi }_{r}\left({C}_{r}\right)={\int }_{{C}_{r}}^{\overline{{C}_{r}}}q\left(\tau \right)d\tau$.

We derive the following by plugging $\lambda =0$ and ${\Pi }_{r}\left({C}_{r}\right)={\int }_{{C}_{r}}^{\overline{{C}_{r}}}q\left(\tau \right)d\tau$ in (A2):

$$\begin{aligned}L&={\int }_{\underset{\_}{{C}_{r}}}^{\overline{{C}_{r}}}\Bigg[\frac{1}{b}\left(a-q\left({C}_{r}\right)\right)q\left({C}_{r}\right)+\left({i}_{r}-{i}_{s}\right)t\left({C}_{r}\right)wq\left({C}_{r}\right)-{C}_{r}q\left({C}_{r}\right)\\ &\quad -{\int }_{{C}_{r}}^{\overline{{C}_{r}}}q\left(\tau \right)d\tau -{C}_{s}q\left({C}_{r}\right)\Bigg]f\left({C}_{r}\right)d{C}_{r}\end{aligned}$$

(A3)

Since $F\left(\underset{\_}{{C}_{r}}\right)=0,F\left(\overline{{C}_{r}}\right)=1$, ${\int }_{\underset{\_}{{C}_{r}}}^{\overline{{C}_{r}}}{\int }_{{C}_{r}}^{\overline{{C}_{r}}}q\left(\tau \right)d\tau \cdot f\left({C}_{r}\right)d{C}_{r}={\int }_{\underset{\_}{{C}_{r}}}^{\overline{{C}_{r}}}q\left(\tau \right)\bullet \frac{F\left({C}_{r}\right)}{f\left({C}_{r}\right)}\bullet f\left({C}_{r}\right)d{C}_{r}$, thus (A3) can be rewritten as:$L={\int }_{\underset{\_}{{C}_{r}}}^{\overline{{C}_{r}}}\Big[\frac{1}{b}\left(a-q\left({C}_{r}\right)\right)q\left({C}_{r}\right)+\left({i}_{r}-{i}_{s}\right)t\left({C}_{r}\right)wq\left({C}_{r}\right)-{C}_{r}q\left({C}_{r}\right)-{C}_{s}q\left({C}_{r}\right)-q\left(\tau \right)\bullet \frac{F\left({C}_{r}\right)}{f\left({C}_{r}\right)}\Big]f\left({C}_{r}\right)d{C}_{r}$, the supplier’s profit is:${\Pi }_{s}=\frac{1}{b}\left(a-q\left({C}_{r}\right)\right)q\left({C}_{r}\right)+\left({i}_{r}-{i}_{s}\right)t\left({C}_{r}\right)wq\left({C}_{r}\right)-{C}_{r}q\left({C}_{r}\right)-{C}_{s}q\left({C}_{r}\right)-q\left({C}_{r}\right)\bullet \frac{F\left({C}_{r}\right)}{f\left({C}_{r}\right)}$(A4). Plugging ${q}^{*}\left({C}_{r}\right)$ into (A4), the second order condition w.r.t. $t\left({C}_{r}\right)$ is: $\frac{{\partial }^{2}{\Pi }_{s}}{\partial {t}^{2}\left({C}_{r}\right)}=\mathrm{b}{i}_{r}{w}^{2}\left(\frac{1}{2}{i}_{r}-{i}_{s}\right)$. When $2{i}_{s}>{i}_{r}$, $\frac{{\partial }^{2}{\Pi }_{s}}{\partial {t}^{2}\left({C}_{r}\right)}<0$, in this case, the trade credit term $t$ is a concave function of the supplier’s profit, i.e., there exists an optimal term of trade credit that maximizes the supplier’s profit. If $2{i}_{s}<{i}_{r}$, then $\frac{{\partial }^{2}{\Pi }_{s}}{\partial {t}^{2}\left({C}_{r}\right)}>0$, in this case, the trade credit term $t$ is a convex function of the supplier’s profit, i.e., there exists an optimal term of trade credit that minimizes the supplier’s profit, the optimal term of trade credit that maximizes the supplier’s profit is either 0 or maximum (or both). If $2{i}_{s}={i}_{r}$, then $\frac{{\partial }^{2}{\Pi }_{s}}{\partial {t}^{2}\left({C}_{r}\right)}=0$. This indicates that the term $t$ of trade credit is a monotonic increasing (or decreasing) function of the supplier’s profit, i.e., the supplier’s profit increases (or decreases) indefinitely with the extension of the trade credit term, which is not in line with practice. Therefore, when $2{i}_{s}>{i}_{r}$, the optimal trade credit term is ${t}^{*}\left({C}_{r}\right)=\frac{1}{\left({{2i}_{s}-i}_{r}\right)w}\left[\frac{\mathrm{a}\left({i}_{r}-{i}_{s}\right)}{\mathrm{b}{i}_{r}}+\frac{{i}_{s}}{{i}_{r}}w+\frac{{{i}_{s}-i}_{r}}{{i}_{r}}{C}_{r}-{C}_{s}-\frac{F\left({C}_{r}\right)}{f\left({C}_{r}\right)}\right]$.

Appendix B: Proof of Proposition 2

For $u={\Pi }_{r1}+{\Pi }_{r2}={\int }_{{C}_{r2}}^{\overline{{\mathrm{C}}_{\mathrm{r}}}}q\left(\tau \right)\mathrm{d}\tau +\left({C}_{r1}-{C}_{r}\right){q}_{1}\left({C}_{r1}\right)+\left({C}_{r2}-{C}_{r}\right){q}_{2}\left({C}_{r2}\right)$, the partial derivative of $u$ w.r.t. ${C}_{r1}$ is: $\frac{\partial u}{\partial {C}_{r1}}={q}_{1}\left({C}_{r1}\right)+\left({C}_{r1}-{C}_{r}\right)\frac{\partial {q}_{1}\left({C}_{r1}\right)}{\partial {C}_{r1}}.$ Referring to Appendix A, we easily derive ${q}_{1}\left({C}_{r1}\right)=\frac{b}{2}[\frac{a}{b}+{i}_{r}{t}_{1}\left({C}_{r1}\right)w-w-{C}_{r}]$, ${t}_{1}\left({C}_{r1}\right)=\frac{1}{\left({{2i}_{s}-i}_{r}\right)w}\left[\frac{\mathrm{a}\left({i}_{r}-{i}_{s}\right)}{\mathrm{b}{i}_{r}}+\frac{{i}_{s}}{{i}_{r}}w+\frac{{{i}_{s}-i}_{r}}{{i}_{r}}{C}_{r1}-{C}_{s}-\frac{F\left({C}_{r1}\right)}{f\left({C}_{r1}\right)}\right]$. Since $\frac{\partial {q}_{1}\left({C}_{r1}\right)}{\partial {C}_{r1}}=\frac{b}{2}{i}_{r}w\frac{\partial {t}_{1}\left({C}_{r1}\right)}{\partial {C}_{r1}}$, $\frac{\partial {t}_{1}\left({C}_{r1}\right)}{\partial {C}_{r1}}=\frac{1}{\left({{2i}_{s}-i}_{r}\right)w}\left[\frac{{{i}_{s}-i}_{r}}{{i}_{r}}-d\left(\frac{F\left({C}_{r1}\right)}{f\left({C}_{r1}\right)}\right)\right]$ where $b>0$, $0<{i}_{s}<{i}_{r}<1,{{2 i}_{s}>i}_{r},d\left(\frac{F\left({C}_{r1}\right)}{f\left({C}_{r1}\right)}\right)\ge 0$, thus $\frac{\partial {t}_{1}\left({C}_{r1}\right)}{\partial {C}_{r1}}<0,$ so $\frac{\partial {q}_{1}\left({C}_{r1}\right)}{\partial {C}_{r1}}<0$. When ${C}_{r1}\le {C}_{r}$, $\frac{\partial u}{\partial {C}_{r1}}$ is strictly larger than zero, thus there is at least one subinterval in $({C}_{r},\overline{{C}_{r}}]$ that fulfills the condition $\frac{\partial u}{\partial {C}_{r1}}>0$, and there is at least one ${C}_{r1}^{\prime}\in ({C}_{r},\overline{{C}_{r}}]$, which indicates the retailer’s profit when disclosing the cost type ${C}_{r1}^{\prime}$ is greater than that when disclosing her true cost type ${C}_{r}$. Hence, the retailer prefers not to reveal her true cost type thus signals a higher cost type.

The partial derivative of $u$ w.r.t. ${C}_{r2}$ is: $\frac{\partial u}{\partial {C}_{r2}}=-{q}_{2}\left({C}_{r2}\right)+{q}_{2}\left({C}_{r2}\right)+\left({C}_{r2}-{C}_{r}\right)\frac{\partial {q}_{2}\left({C}_{r2}\right)}{\partial {C}_{r2}}=\left({C}_{r2}-{C}_{r}\right)\frac{\partial {q}_{2}\left({C}_{r2}\right)}{\partial {C}_{r2}}$. Analogously, $\frac{\partial {q}_{2}\left({C}_{r2}\right)}{\partial {C}_{r2}}<0$. When $\underset{\_}{{C}_{r}}<{C}_{r2}<{C}_{r}$, $\frac{\partial u}{\partial {C}_{r2}}>0$, $u$ is increasing in ${C}_{r2}$; when ${C}_{r2}={C}_{r}$, $\frac{\partial u}{\partial {C}_{r2}}=0$; when ${C}_{r}<{C}_{r2}<\overline{{C}_{r}}$, $\frac{\partial u}{\partial {C}_{r2}}<0$, $u$ is decreasing in ${C}_{r2}$; thus $u$ is maximized when ${C}_{r2}={C}_{r}$. That is, the retailer could secure a higher profit by revealing her true cost information.