Managing Inventory for Entrepreneurial Firms with Trade Credit and Payment Defaults

Transcription

1 Managing Inventory for Entrepreneurial Firms with Trade Credit and Payment Defaults Wei Luo Kevin Shang IESE Business School, University of Navarra, Av. Pearson 21, Barcelona, Spain Fuqua School of Business, Duke University, Durham, North Carolina 27708, USA September 25, 2013 This paper considers an entrepreneurial firm that periodically orders inventory to satisfy nondecreasing demand in a finite horizon. The firm provides trade credit to its customer while receiving it from its supplier. A default penalty cost is incurred for unfulfilled payments. The objective is to obtain an inventory policy that minimizes the total inventory and default penalty cost. Utilizing an accounting concept of working capital (which incorporates cash, inventory, accounts receivable and accounts payable), we prove that a myopic policy is optimal when the sales collection period is longer than or equal to the purchases payment period. The myopic policy has a simple structure an order is placed to achieve a target base-stock level that depends on the firm s working capital. When the payment period is longer than the collection period, we derive a lower bound to the optimal cost and propose an effective heuristic that has a generalized form of the above structured policy. These policies not only generalize the classic base-stock policy, but also resemble practical working capital management under which firms make inventory decisions according to their working capital status. The policy parameters have a closed-form expression, which shows the impact of demand and cost parameters on the inventory decision. Our study assesses the value of considering financial flows when the firm makes the inventory decision and reveals insights that are consistent with empirical findings in the literature. Finally, we demonstrate how supplier s liquidity provisions can mitigate the distortion of demand information. Key words: trade credit, inventory policy, payment default, bullwhip effect

2 1 Introduction Small and medium-sized enterprises (SMEs) account for 95% to 99% of registered business depending on the country. The contribution of SMEs to GDP ranges from less than 20% in poor countries to about half of GDP in rich economies (International Trade Centre 2009). Trade finance is the lifeline for small or entrepreneurial firms as they are newly established and usually have difficulty securing bank loans. This is particularly true during the financial crisis when banks are extremely risk-averse. Thus, it is very important to investigate how entrepreneurial firms should manage their working capital when trade credits are present in their business transactions. We consider an entrepreneurial firm in the middle of a supply chain. The firm periodically orders inventory from its supplier to fulfill stochastic and non-decreasing demand received from its customer in a finite horizon. The firm has no credit history to secure bank loans, so trade credit is the single source of external financing. Specifically, the firm grants trade credit to its customer while receiving it from its supplier. The trade credit is a one-part (net term) contract, that is, the payment is due within a certain time period after the invoice is issued. Thus, the firm pays for the ordered inventory after a deferral payment period following the delivery of goods, and receives sales revenue after a collection period following the demand. As suggested by the 1998 National Survey of Small Business Finances (NSSBF) data, a big portion of the small firms declared that they had made some payments to their suppliers after the due date of the trade credit. These post due-date payments, referred to as payment defaults, often incur little monetary penalty for the buyers (see discussions in 2.4.2). Nevertheless, payment default will hurt a firm s credit record, making it hard to finance in the future 1. In light of this, we introduce a default penalty cost incurred upon the unfulfilled payment to the firm s supplier. A general question we study in this work is how should the entrepreneurial firm manage its inventory so as to minimize its total cost (consisting of the inventory purchasing cost, holding and backorder costs, as well as the default penalty cost)? Our research question is related to working capital management for entrepreneurial firms. Working capital refers to the difference between current assets and current liabilities (short-term assets and liabilities with maturities of less than one year). On the balance sheet, current assets include cash, inventory, and accounts receivable (A/R). Current liabilities include accounts payable (A/P) and short-term loans 2. With trade credits, inventory decisions directly affect working capital levels as the deferred inventory payment and the delayed sales collection are recorded as A/P and A/R, respectively. The goal of working capital management is to increase the profitability of a firm through investing in long-term, high-return assets while ensuring that it has sufficient liquidity to 1 For example, Dun and Bradstreet keeps credit records and provides credit reports of small businesses. 2 We do not consider short-term loans due to the lack of bank financing assumed in this model. 2

3 meet short-term operations, e.g, paying suppliers for inventory in our context (Pass and Pike 1984). There is an inherent tradeoff between profitability and liquidity: keeping too much cash for operations will lead to a smaller return from the long-term investment. In practice, firms usually decide on a working capital policy, which budgets how much sales revenue to be retained for working capital. Typically, three types of capital budgeting strategies are considered: aggressive, moderate, and conservative (Gallagher and Andrew 2007, p ). With an aggressive strategy, firms choose to keep a smaller portion of sales revenue as working capital (which increases the risk of illiquidity). In this paper, we assume a given working capital policy and study the optimal inventory policy when trade credits are present for both upstream and downstream transactions. Most inventory models in the literature do not explicitly consider the interrelationship between inventory decisions and cash flows under the assumption of an efficient market, i.e., external funding is available without market friction. Thus, financial decisions can be decoupled from the operational decisions (Modigliani and Miller 1958). We believe that it is important to study this interrelationship for the following two reasons. First, since the financial crisis in 2008, it has been extremely difficult for entrepreneurial firms to secure bank financing and meet their short-term operations. Due to these market frictions, the inventory decision plays a significant role on a firm s liquidity and operational efficiency. More specifically, today s inventory order decision will directly affect its future cash payment. If a firm orders too much, it will not only incur a higher holding cost, but also increase the chance of future payment default. On the other hand, if a firm orders too little, it is more likely to incur a higher backorder cost. Thus, there is a clear tradeoff between these system costs when making an inventory decision. Second, for the traditional inventory problem, there exist very simple yet powerful models (e.g., Newsvendor, base-stock, etc.) that illustrate how inventory decision is affected by the system parameters. Given that the financial flow cannot be ignored from the inventory decision, the practitioners advocate the importance of interdisciplinary study between operations and accounting/finance (e.g., working capital management, supply chain finance, etc.). Yet, there are no simple models and frameworks that can help managers understand their relationships. We hope to fill this gap through this study. We formulate the inventory system with trade credit into a multi-state dynamic program that keeps track of inventory level, cash balance, as well as different ages of A/P and A/R within the payment and collection periods, respectively. This dynamic program is hard to solve because the state space is high-dimensional. We utilize the accounting concept of working capital (= cash + inventory + accounts receivable - accounts payable) to redefine the state and simplify the original dynamic program. We consider three cases regarding the different lengths of payment and collection periods. When the payment period is equal to or less than the collection period, we prove that a 3

4 myopic policy is optimal. The optimal policy is operated under two control parameters (d, S), d S: the firm reviews its inventory-equivalent working capital level (i.e., working capital divided by the unit purchase cost) and inventory position at the beginning of each period; if the working capital is lower (resp., higher) than the default threshold level d (resp., the base-stock level S), the firm places an order to bring its inventory position up to d (resp., S); if the working capital level is between d and S, the firm orders up to the working capital level. When the payment period is longer than the collection period, the firm s payment beyond the collection period depends on the future cash inflow, which in turn depends on the random demand. Consequently, it is difficult to characterize the optimal policy. Nevertheless, we develop a lower bound to the optimal cost and propose an effective heuristic. The heuristic policy, referred to as the (d, a, S) policy, is operated under five control parameters and can be viewed as a generalization of the (d, S) policy. In a numerical study, we show that the heuristic is near optimal. The optimal and heuristic policy parameters have a closedform, Newsvendor-type expression which allows us to investigate the impact of demand variability on the inventory decision as well as the tradeoffs between inventory holding cost, backorder cost, and payment default cost. Consequently, we can use them to gain insights. Finally, we also test our optimal and heuristic policies under different non-stationary demand forms, and the performance remains satisfactory. Thus, the suggested policies can be comfortably applied to systems with general demand patterns. These policies resemble practical working capital management under which firms make inventory decisions according to the working capital level (Aberdeen Group 2009). We summarize the main contributions. First, managers are hindered from integrating accounts payable/receivable into the inventory policy due to the typical organizational structure of the firm (i.e., the former is a function of a treasurer, and the latter an operations manager). These two functions need to be aligned when the financial market is not efficient. We present a model that captures the dynamics between inventory decisions and accounts payable/receivable resulted from trade credit terms, and provide simple and implementable inventory policies. These policies suggest that a firm should consider the entire working capital and the default penalty when making inventory decisions. This result augments the current scope of operations by incorporating accounting concepts. We also assess the value of considering financial flows when making inventory decisions in a numerical study. Second, the optimal policy generalizes the classic base-stock policy (or the Newsvendor solution) and the cash-constrained base-stock policy (i.e., the base-stock level generated by treating cash as a hard constraint) proposed by Bendavid et al. (2012). Specifically, when the system working capital is high or the default penalty is zero (i.e., cash is not a concern), the optimal policy reduces to the classic base-stock solution; on the other hand, when the default penalty is very large (infinity), our 4

5 policy suggests the firm order up to the working capital level, which is equivalent to the solution obtained from the cash-constrained model. We believe that the real-world practice falls between these two extremes, and our model reflects this generality. In addition, the optimal policy has an appealing closed-form expression, which facilitates classroom teaching and provides clear intuition for the impact of system parameters on the optimal policy. Third, our model reveals several insights that are consistent with empirical findings in the finance literature. For example, our policy indicates that a firm would possibly choose to default on the payment if its current working capital level is low and the backorder penalty is higher than the default penalty. This explains why payment defaults are commonly observed in practice as the default penalty cost is usually small. In addition, we show that it is more beneficial for the firm to extend the payment period with its supplier if the firm also grants trade credit to its customer. This finding explains why there exists a positive correlation between the firm s upstream and downstream credit periods (Fabbri and Klapper 2009). Through our model, we also predict that customer payment default could lead to the distortion of demand information (either bullwhip or reverse bullwhip effects, see Lee at al. 1997, Rong et al. 2008). We demonstrate that the supplier could effectively mitigate this distortion through liquidity provision. This complements Boissay and Gropp s (2007) finding that the downstream liquidity shocks can be absorbed by the suppliers through liquidity provision. Lastly, a firm often wishes to extend the payment period and shorten the collection period to reduce its cash conversion cycle (cash collection periods + on-hand inventory in periods - inventory payment periods). We can quantify this benefit through our model. However, extending the payment period may induce the supplier to increase the unit wholesale price, which may not benefit the buyer. Thus, we generate an isocost curve for different pairs of purchase price and payment period, which provides guidance for firms to offer an early payment in exchange for a lower purchase price. 2 Literature Review and Model Assumptions Our paper is related to a few research topics summarized below. In particular, in 2.4 we shall discuss several empirical findings, which set the stage for our model assumptions. 2.1 Inventory Models with Trade Credit This literature can be categorized based on whether a single- or multi-period problem is considered. For the single-period model, Zhou and Groenevelt (2008) consider the impact of financial collaboration in a third-party supply chain. They find that the total supply chain profit with bank financing is slightly higher than that with open account (trade credit) financing. Kouvelis and Zhao 5

6 (2009) consider capital-constrained retailer and supplier. They show that a risk-neutral supplier should always finance the retailer at rates less than or equal to the risk-free rate. Yang and Birge (2010, 2012) study how different priority rules of order repayment influence trade credit usage. They consider bankruptcy default, which is different from payment (illiquidity) default assumed in our model. As for the multi-period model, this literature can be further categorized based on how trade credits are modeled. One category is to characterize the impact of trade credit on the inventory holding cost. This literature implicitly assumes that cash is always available so cash dynamics are not explicitly modeled. Beranek (1967) uses a lot-size model to illustrate how a firm s inventory holding cost should be adjusted according to the firm s actual financial arrangements. Maddah et al. (2004) investigate the effect of permissible delay in payments on ordering policies in a periodicreview (s, S) inventory model with stochastic demand. They develop approximations for the policy parameters. Gupta and Wang (2009) consider a stochastic inventory system where trade credit term is modeled as a non-decreasing holding cost rate according to an item s shelf age. Under the assumption that the full payment is made when the item is sold, they prove that a base-stock policy is optimal. Huh et al. (2011) and Federgruen and Wang (2010) generalize the results of Gupta and Wang. Song and Tong (2012) consider an inventory system where a base-stock policy is implemented. They investigate how the holding cost rate is affected by the different payment and collection time epochs. Another category, which is more related to our model, is to explicitly characterize cash flow dynamics resulted from the trade credit terms. Haley and Higgins (1973) expand Beranek s model and consider the problem of jointly optimizing inventory decision and payment times when demand is deterministic and inventory is financed with trade credit. Schiff and Lieber (1974) consider the problem of optimizing inventory and trade credit policy for a firm where the demand is deterministic but depends on the credit term and inventory level. Bendavid et al. (2012) study a firm whose replenishment decisions are constrained by the working capital requirement. Their model is similar to ours in that they also consider how inventory replenishment is affected by the payment and the collection periods. However, their model considers i.i.d demand and implements a base-stock policy with inventory ordering subject to a hard constraint on the amount of working capital. Thus, no defaults are allowed. They characterize the dynamics of system variables and obtain the optimal base-stock level via a simulation approach. Modeling cash flows is necessary in these models because cash is a system state subject to financial constraints or penalties, which in turn affect operational decisions. 6

7 2.2 Interface of Operations and Finance There has been an emerging research stream that aims to study problems that involve both operations and finance without considering trade credit. Xu and Birge (2004) analyze the interactions between a firm s production and financing decisions as a tradeoff between the tax benefits of debt and financial distress costs. Xu and Birge (2006) propose an integrated corporate planning model, which extends the forecasting-based discount dividend pricing method into an optimization-based valuation framework to make production and financial decisions simultaneously for a firm facing market uncertainty. Chao et al. (2008) consider a self-financed retailer who replenishes inventory in a finite horizon. Tanrisever et al. (2012) explore the tradeoff between investment in process development and reservation of cash in order to avoid bankruptcy for a start-up firm. They provide managerial insights by characterizing how to create operational hedges against the bankruptcy risk. Boyabatli et al. (2013) study the impact of budget constraints on the technology choice. They show that ignoring financial constraints may lead to mis-specifications. Li et al. (2013) study a dynamic model in which inventory and financial decisions are made simultaneously in order to maximize the firm s value the expected present value of dividends minus total capital subscriptions. Luo and Shang (2013) integrate material and cash flow in a supply chain. They characterize the optimal joint inventory and investment policy and investigate the value of cash pooling. Other noteworthy examples include Babich and Sobel (2004), Buzacott and Zhang (2004), Ding et al. (2007), Dada and Hu (2008), Caldentey and Chen (2010). The research questions in these papers are quite different from ours. 2.3 Inventory Models with Capacity and Advanced Demand Information Our model is related to two streams of inventory problems. The (inventory-equivalent) working capital level serves as an upper bound for the inventory order up-to level when it is larger than the default threshold d under the (d, S) policy. In this regard, our model is related to the capacitated inventory model. We refer the reader to Tayur (1997) for a review and Levi et al. (2008) for recent developments. The other stream is inventory systems with advance demand information. The incoming and outgoing cash flows in accounts receivable and payable pipelines can be viewed as advanced cash flow information. For this research stream, see Özer and Wei (2004) and references therein. 2.4 Finance Literature on Trade Credit The motivation and several key assumptions of our model are based on the following empirical finance literature. 7

8 2.4.1 One-Part Trade Credit Trade credit is widely used for business transactions in supply chains, and is the single most important source of external finance for firms (Petersen and Rajan 1997). It appears on the firm s balance sheet and accounts for about one half of the short-term debt in two samples of UK and US firms (Cuñat 2007). In the finance literature, there have been various theories, such as order incentives (Schwartz 1974), taxes (Brick and Fung 1984), transaction costs (Ferries 1981), and information asymmetries (Smith 1987, Lee and Stowe 1993) that explain the existence of trade credit; see Cuñat and Garcia-Appendini (2012) for an excellent review. Trade credit is one of the key sources of funding for small, entrepreneurial firms that lack collateral and credit history. It is well documented that trade credit is more common among newly created firms and those with less tangible assets (Berger and Udell 1998, Elliehausen and Wolken 1993). Among various types of trade credit contracts, the one-part (net term) is the simplest form. Cuñat (2007) indicates that there is a big portion of firms that use one-part contracts 3. Our paper takes one-part trade credit as a premise and aims to investigate its impact on a firm s inventory policy and the resulting operating cost. Finally, two-level trade credits are commonly seen in practice, i.e., firms often provide trade credit to its customer while receiving it from its supplier. Fabbri and Klapper (2009) find evidence of a positive correlation between a firm s upstream and downstream credit period length. Guedes and Mateus (2009) examine trade credit linkages on the propagation of liquidity shocks in supply chains. These papers motivate us to study two-level trade credits and our numerical results also echo their findings Late Payment and Default Penalty Cost According to Table 1 in Cuñat and Garcia-Appendini (2012), late payments are not uncommon In the category of net term days 4, 17.2% of the firms make payment after the due date. The same table also shows that the cost of late payment is minimal. Specifically, the average surcharge for each delayed dollar over the time after the due date is about 0.92 cents in the same category. In fact, many suppliers do not charge an explicit penalty for late payment. This distinctive feature shows how trade credit repayment contains an extra degree of flexibility, which can be extremely useful for small and entrepreneurial firms that have more volatile sales and are financially more fragile. One reason that the supplier does not charge for the payment default is the on-going relationships with the buyer (Cuñat 2007). Wilson and Summers (2002) provide rationale of sustaining this long-term business relationship. These findings support our model assumption that the supplier continues 3 According to the 1998 NSSBF survey, 49% of the trade credit contracts are one-part. 4 Net days represents the most common one-part trade credit used by the firms surveyed in the 1998 SSBF. 8

9 to do business with the firm which may occasionally default on the payment. Although the late payment causes little monetary penalty to the buyer, it is necessary to be factored in when making inventory decisions as frequent default behavior will hurt a firm s credit record, making bank finance or future transactions difficult in a later stage (Cook 1999; Garcia-Appendini 2007). In light of this, we introduce a default penalty cost into the model, which can be viewed as a backorder penalty cost charged on the unfilled payment Fixed Payment Period Our model assumes that once the firm and its business partners have agreed upon a net term contract, the payment periods are fixed within the horizon. This is consistent with an empirical finding in Ng et al. (1999), where the trade credit terms (payment period in our context) may be different across industries, but they are relatively stable within each industry and along time. For example, the authors find that net 30 (i.e., pay in full within 30 days) is the most common net term contracts. Nevertheless, credit policy is an organizational design choice and firms have incentives to offer an early payment (say, pay-on-shipment) in exchange for a lower purchase price before agreeing on the trade credit contract. Our model provides a clear tradeoff between the purchase price and the payment period, which can be used to assist in negotiation Working Capital Policy Firms in different industries have various standards in terms of determining the requirement for the working capital. In general, a firm can adopt three types of working capital strategies: aggressive, moderate, and conservative (Gallagher and Andrew 2007, p ). With an aggressive strategy, firms choose to retain a smaller portion of sales revenue for working capital and invest more on the long-term assets with higher return. However, budgeting too little working capital will increase the risk of illiqudity (i.e., payment defaults in our context). In practice, the working capital budgeting decision is usually made in a firm s annual Sales and Operations Planning (S&OP) meeting, which is a medium-term strategy (12 to 18 months). In this paper, since our focus is the short-term operational decision, we assume a hierarchical decision process in the sense that the medium-term working capital strategy has been set to the model. The rest of this paper is organized as follows. 3 describes the model and formulates the corresponding dynamic program. 4 focuses on the model with balanced payment and collection periods and proves the optimal policy. 5 ( 6) considers the model with longer payment (collection) periods. 7 discusses the qualitative insights through numerical studies. 8 concludes. Appendix A shows the optimality of the (d, a, S) policy. Appendix B examines the effectiveness of the heuristics. Ap- 9

10 pendix C provides all proofs. Throughout this paper, we define x + = max(x, 0), x = min(x, 0), a b = max(a, b), and a b = min(a, b). 3 The Model We consider a finite-horizon, periodic-review inventory system where a firm orders from its supplier and sells to its customer. A one-part trade credit is employed for transactions at both upstream and downstream, that is, the firm pays its supplier after a payment period following the delivery of goods, and receives cash from its customer after a collection period following the demand. In accounting, the inventory payment period (sales collection period) is also referred to as the payables (receivables) conversion period or days purchases (sales) outstanding. The payment and collection periods jointly affect the cash conversion cycle (CCC), which is defined as CCC = Inventory conversion period + Collection period Payment period. Transactions based on trade credit will affect a firm s accounts payable (A/P) and accounts receivable (A/R). Table 1 lists the four events and the corresponding changes in inventory and cash flow, as well as accounts payable and receivable. Transaction Inventory/Cash flow Accounting Variables Receiving X units of inventory Inventory X A/P $X Selling Y units of inventory Inventory Y A/R $Y Paying $X to the supplier Cash $X A/P $X Collecting $Y from the customer Cash $Y A/R $Y Table 1: Events and accounting variables associated with CCC We now formalize the above description into our model. Since the focus is on cash and inventory dynamics under trade credit, for simplicity and without loss of generality, we assume that the shipping lead time is zero. Let m be the payment period and n be the collection period. We count the time forward, i.e., t = 0, 1, 2, etc. The sequence of events is as follows: At the beginning of period t, (1) inventory order decision is made; (2) shipment arrives; (3) payment due in this period (corresponding to the inventory ordered in period t m) is made to the supplier; (4) a default penalty cost is incurred in case of insufficient payment; demand is realized during the period and customer payment due in this period (corresponding to the sales in period t n) is collected; at the end of the period, all inventory related costs are calculated. The objective is to minimize the firm s total discounted cost over the entire horizon of T periods. Customer demand in period t is modeled as a nonnegative random variable D t with probability 10

11 density function (p.d.f.) f t, cumulative distribution function (c.d.f.) F t, mean µ t and variance σt 2. The demand is independent and stochastically non-decreasing from period to period. This represents the growth of the market faced by the entrepreneurial firm. (We, however, also test our results for general non-stationary demand; see Appendix B2.) We assume that the unsatisfied demand is fully backlogged. Define the state and decision variables at the beginning of period t: z t = order quantity made in Event (1); x t = net inventory level before Event (2); w t = net cash level before Event (3); P t = (P t m,..., P t 1 ): m-dimensional vector of accounts payable; R t = (R t n,..., R t 1 ): n-dimensional vector of accounts receivable. Here, P t i and R t j denote the A/P and A/R created in period t i and t j, respectively, for i = 0, 1,..., m and j = 0, 1,..., n. So P t m and R t n are the most aged A/P and A/R, while P t and R t are A/P and A/R created in the current period. Figure 1 shows these system variables in period t with the material and cash flows in solid and dashed arrows, respectively. Figure 1: The base model with material flow and cash flow We introduce the inventory cost parameters. Denote h as the holding cost per unit inventory per period, and b the backorder cost per unit backorder per period. If the inventory position at the beginning of the period is y, then the holding and backorder cost of the period can be expressed as H t (y) = E[h(y D t ) + + b(y D t ) ]. Here and in the sequel, the expectation is taken over D t, unless otherwise specified. Let p denote the default penalty per dollar per period 5. The single-period cost function is Ĝ t (x t, z t, w t, P t m ) = H t (x t + z t ) + p (w t P t m ) + α m cz t, (1) 5 Here we focus on the illiquidity default, which is different from the bankruptcy default studied in Yang and Birge (2012, 2012). To model the latter, one can set the penalty rate p to a large value, making on-hand cash a hard constraint similar to Bendavid et al. (2012). 11

12 where the first term represents the expected inventory holding and backorder cost; the second term is the default penalty cost for not being able to make the payment in full at the end of the payment period; the last term is the procurement cost, which is realized in m periods later when payment is due. We charge this cost in the current period. Denote P i t as the vector consisting of the first i elements of P t, and P i t as vector P t without the first i elements. Let r be the unit revenue retained for operations under the firm s working capital policy 6 and c the unit procurement cost. Here we assume that r is a pre-determined parameter, which is an output of the firm s medium-term S&OP meeting. According to Table 1, the dynamics of states between two periods are: x t+1 = x t + z t D t, (2) w t+1 = w t P t m + R t n, (3) P t+1 = (P 1 t, cz t ), (4) R t+1 = (R 1 t, rd t ). (5) As shown In the cash dynamics (3) and (5), we start with the model where firm is guaranteed to receive a full payment from the customer in n periods after the trade. We shall continue with the model extension where customer default is considered; see 7.3. It is worth noting that, although we include inventory holding, backorder and default penalty costs in the periodic cost function in (1), these costs do not appear in the cash dynamics in (3). The reasons are the following. Firstly, we assume that the physical holding cost is not reflected in the periodic cash outflows due to accrual accounting. (The accrual method measures the performance of a company by recognizing economic events regardless of when cash transactions occur.) Secondly, the backorder cost is considered as a non-monetary penalty cost (e.g., loss of goodwill) in our model. Lastly, according to the finance literature described in 2.4.2, the default penalty includes two parts: a monetary part equal to the interest charged by the supplier upon overdue payment and a nonmonetary part representing the intangible consequence of defaults, such as loss of credibility. The first part is usually very small, if any, for the one-part trade credit contract, so we omit it from consideration. As stated, the second part is similar to the backorder cost assumed in the classic inventory model. The intangible cost of backorder (resp., default penalty) represents firm s target service level on customer order (resp., supplier payment) fulfillment. Denote ˆV t (x t, w t, P t, R t ) as the minimum expected cost over period t to T + 1, and over all 6 For example, if sales price is $5 and r = $2, this means that 40% of the revenue is retained as working capital. A larger r can be interpreted as a more conservative working capital strategy described in

13 feasible decisions. Let α be the single-period discount rate. The dynamic program is { ˆV t (x t, w t, Ĝt (x P t, R t ) = min t, z t, w t, P t m ) 0 z t +αe ˆV t+1 (x t+1, w t+1, P t+1, R t+1 ), m+1 ˆV T +1 (x T +1, w T +1, P T +1, R T +1 ) = α m cx T +1 + α s 1 Ep (w T +s P T +s m ), (7) s=1 } (6) where the expressions of w T +s follow the dynamics shown in (3). Here, we make two assumptions for the terminal cost function. First, we assume that the end-ofhorizon inventory has the unit salvage value c, and backlogged demand has to be satisfied. This can be interpreted by setting z T +1 = x T +1. That is, the supplier will buy back the left-over inventory x + T +1 at the unit price c or the firm has to make a final order of x T +1 to fulfill the unsatisfied demand. Correspondingly, we have P T +1 = cx T +1. This payment is realized after m periods, and we charge the resulting cost α m cx T +1 to period T + 1, which explains the term in (7). Second, we assume that the default penalty applies from time T + 1 to T + m + 1. When m > n, the derivation of w T +s (s = 1,..., m + 1) needs the additional sales from the next planning horizon. This explains the second part of the terminal cost, in which the expectation is taken over the sum of demands that are necessary to derive the net cash level w T +s. The base model formulated in (6) and (7) is difficult to solve. One can show the joint convexity of ˆV t ( ) and Ĝt( ), thus deriving a state-dependent global optimal policy. However, computing this policy is extremely hard due to the curse of dimensionality (state space has m + n + 2 dimensions). In the next three sections, we introduce new system variables to reduce the state space, and provide optimal solution or simple heuristics for the base model. 4 Balanced Credit Periods This section considers the system with equal payment and collection periods, i.e., m = n = λ. For simplicity, we refer to this as the λ-model. We shall prove that this model can be solved by redefining a new state variable. 4.1 State Space Reduction When the payment period equals the collection period, the firm knows exactly how much it should pay the supplier and receive from the customer for the next λ periods according to the known pipeline P t and R t, respectively. Thus, the firm has complete information about cash dynamics in the incoming λ periods. We now reduce the state space by introducing new system variables. Let 13

14 e λ be the λ-dimensional column vector of ones. Define y = x + z, w = x + (w Pe λ + Re λ )/c. We refer to y as the inventory position and w as the working capital level measured in inventory units, at the beginning of period t. This is consistent with the accounting definition of net working capital, which equals to current assets (inventory, cash, and A/R) minus current liabilities (A/P). Let p = cp, ρ = r/c, and θ = ρ 1. Applying dynamics (2)-(5) repeatedly, we have ˆV t (x t, w t, P t, R t ) = p (w t P t λ ) + αp (w t P t λ + R t λ P t λ+1 ) + + α λ 1 p (w t P t λ P t 2 + R t λ + + R t 2 P t 1 ) + V t (x t, x t + (w t P t e λ + R t e λ )/c), (8) and that V t satisfies the functional equation V t (x, w) = min x y {G t(x, w, y) + αev t+1 (y D t, w + θd t )}, (9) where the one-period cost function can be shown as G t (x, w, y) = H t (y) + α λ p(y w) + + α λ c(y x). (10) Since z T +1 = x T +1 is equivalent to y T +1 = 0, the terminal cost function becomes V T +1 (x, w) = α λ cx + α λ pw. (11) As shown in (8), the default penalty costs incurred from period t to t + λ 1 can be viewed as sunk costs, which do not affect the inventory decision in period t. Therefore, we charge the default penalty cost occurred in period t + λ to the current period t. This cost shifting scheme enables us to replace the entire state space with only two state variables x t and w t, without losing any useful information for making the optimal inventory decision. To see the transformation of the default penalty term, notice that p (w t P t e λ + R t e λ P t ) = p (cx t + w t P t e λ + R t e λ cx t cz t ) = p(y t w t ) +. Note in (9) that when θ > 0, the firm s working capital will grow with demand. This does not necessarily suggest that overtime working capital becomes irrelevant because the inventory order size also grows as demand increases. In the classic single-stage inventory problem, Karlin and Scarf (1958) introduce the notion of inventory position that transforms the original multi-state dynamic program into a single-state 14

15 problem and proves that a base-stock policy is optimal. Here, we derive a similar result for the inventory model with two-level trade credit contracts. More specifically, by introducing the notation of working capital w, we show that the optimal inventory decision can be determined by the transformed dynamic program in (9)-(11), referred to as the transformed λ-model. Nevertheless, this model is more complicated than the classic inventory problem as it has two state variables: working capital w and inventory level x. The complexity comes from the fact that the inventory decision y in the single-period cost function G t depends on both w and x. In the next subsection we proceed to show how to derive the optimal policy by further decoupling the states. 4.2 The Optimal Policy We shall show that a myopic policy is optimal for the dynamic problem in (9)-(11). This myopic policy has a simple structure that can be implemented easily and reveals insights. As we shall see in Appendix B2, the myopic policy remains very effective for the general demand case. We first explain the myopic policy, which includes two control parameters (d, S) in each period. The policy is operated as follows: the firm monitors its inventory level x and working capital w at the beginning of each period. If w d, the firm orders inventory up to d; if d < w S, the firm uses up all cash and orders inventory up to w; if w > S, the firm orders inventory up to S. Denoting y as the resulting optimal base-stock level, the (d, S) policy can be mathematically stated as y (w) = (d w) S, (12) and graphically illustrated in Figure 2(b). Figure 2: The optimal solution of the transformed λ-model We next illustrate how these optimal control parameters are obtained. For fixed w, the uncon- 15

16 strained myopic minimization problem at period t can be written as { v t (w) = min g t (y) + α λ p(y w) +}, (13) y where g t (y) = H t (y) + α λ (1 α)cy. Here, H t (y) + α λ cy is the single-period purchase cost and inventory related cost; α λ+1 cy represents the fact that the myopic system allows returns with a refund of c for any unsold unit. Let S t be the optimal base-stock level without considering the default penalty cost, i.e., { } S t = arg min g t (y). (14) y This happens when cash is ample or p = 0, making p(y w) + = 0. We term S t the ample-cash base-stock level. We define the default threshold as follows: { } d t = sup y : y g t(y) α λ p. (15) To solve the problem in (13), we consider three cases. (See Figure 2(a).) Case 1. When w d t, the system s working capital is lower than the default threshold d t. In this case, the firm has an incentive to order up to d t as the marginal backorder cost outweighs the marginal holding and default penalty cost. Thus, we have v t (w) = L t (w) = α λ p(w d t ) + g t (d t ). Case 2. When d t < w S t, the system is constrained by the working capital. Now it is optimal to order up to w as ordering either less or more will lead to a higher cost than g t (w). v t (w) = g t (w). Thus, Case 3. When S t < w, the system has ample working capital and orders up to the target base-stock S t. In this case, there is extra cash left after ordering, and v t (w) = g t (S t ). We summarize the above three cases into the following proposition. Proposition 1. The (d, S) policy in (12) is optimal for the myopic problem in (13). As a result, Equation (13) becomes L t (w), v t (w) = g t (w), g t (S t ), if w d t if d t < w S t, (16) if S t < w and the critical ratios of the control parameters can be found in the following lemma. Lemma 1. The control parameters of the myopic policy in (12) satisfy F t (d t ) = b αλ p α λ (1 α)c h + b 16, F t (S t ) = b αλ (1 α)c. (17) h + b

17 The firm will have a chance to default when d t >. From (17), this scenario happens when p α λ b (1 α)c. When α = 1, this condition becomes p b. This explains why payment default is prevalent in practice as the penalty cost is often very small (see 2.4.2). Conversely, the firm will not default as long as p > b. In general, when the default penalty is sufficiently large, the firm will not default, so the resulting model is similar to the cash-constrained model (i.e., cash becomes a hard constraint that restricts the inventory decision) studied by Bendavid et al. (2012). On the other hand, when there is ample cash supply, the working capital can always be set equal to the ideal level S t. In this case, the (d, S) policy is degenerated into the classic base-stock policy, where the base-stock level is exactly the same as S t shown in (17). To facilitate the subsequent discussion, we define the band at time t as B t = { (x t, w t ) R 2 x t yt (w t ) }. (18) This band establishes the region where inventory does not exceed the base-stock level. Figure 2(b) depicts the piecewise linear function y (w). By definition, the band B covers the area below y (w) on the x-w plain. As illustrated in Figure 2(b), the band is divided into three sub-regions. When w d, the firm falls into the default region where the optimal order policy will lead to negative cash and late payment; when d w < S, the firm will hold zero cash after ordering, i.e., cash working capital constraint is binding; when S w, the firm has sufficient cash and orders up to the default-free base-stock. Consequently, the working capital constraint is non-binding. To formally characterize the firm s order strategy under default risk, we define the optimal default quantity as u (w) = (y (w) w) +. Figure 2(b) implies that u (w) is decreasing in w, meaning that the firm will default less if there is more working capital. This optimal behavior echoes the empirical findings that the operational decisions of smaller firms are more aggressive and thus induce higher default risks. We finally show that the (d, S) policy is indeed optimal for the transformed λ-model (or equivalently, the myopic solution is indeed optimal). Proposition 2. (a) The control parameters d t and S t are non-decreasing in t and d t S t for all t; (b) V t (x, w) = α λ cx + W t (w) for all t and (x, w) B t, where W t (w) = v t (w) + αew t+1 (w + θd t ), and W T +1 (w) = α λ pw ; W t (w) is convex in w; (c) The (d, S) policy is optimal for the transformed λ-model. 17

18 Proposition 2(a) implies that if the initial state (x, w) falls in the band, the system states at the beginning of each period will remain in the band under the (d, S) policy. This property ensures the optimality of the myopic policy, which is similar to the classic inventory model. 5 Longer Payment Period (m > n) When the payment period is longer than the collection period, i.e., m > n = λ, the firm has complete cash outflow information for the next m periods and cash inflow information for the next λ periods. Consequently, the cash inflow from period t + λ + 1 to t + m is uncertain for the firm standing at time t. As we shall see, this uncertainty leads to the complication for the analysis. We term this model as the m-model. Recall the base model in Equations (6) and (7). We conduct a similar analysis as in 4 to derive the transformed dynamic program. Keeping the same notation as before without confusion, we define the working capital level measured in inventory units as w = x + (w Pe m + Re λ )/c. In addition, we define the aggregated demand as D m t = D t D t+m 1 and D 0 t = 0. Let F m t, f m t, µ m t, and (σ m t ) 2 be the c.d.f., the p.d.f., mean, and variance of the random variable of D m t, respectively. Moreover, denote F m and ˆF m as the complementary cumulative distribution function (c.c.d.f.) and the loss function of random variable D m. That is, ˆF m (x) = x F m (y)dy. With a similar analysis as in 4.1, the optimal inventory policy for the base model in (6)-(7) can be determined by the following transformed dynamic program: where the single-period cost function is V t (x, w) = min x y {G t(x, w, y) + αev t+1 (y D t, w + θd t )}, (19) G t (x, w, y) = H t (y) + α m Ep(y w ρd m λ t+λ )+ + α m c(y x), (20) and the expectation is taken over D m λ t+λ. The terminal cost function is modified to V T +1 (x, w) = α m cx + α m Ep(w + ρd m λ T +1+λ ). (21) Notice that λ is the number of periods of the known cash flow and will not affect the policy structure. Thus, for ease of exposition, we shall omit λ and reformulate the model with λ = 0. (In our numerical study, however, we still resume λ for calculating the policy and the corresponding 18

19 cost.) Let u = y w, and the default penalty cost can be rewritten as M t (u) = E D m t p(u ρd m t ) +. (22) And the dynamic program becomes V t (x, w) = min x y {H t(y) + α m M t (y w) + α m c(y x) + αev t+1 (y D t, w + θd t )}, (23) V T +1 (x, w) = α m cx + α m E D m T +1 p(w + ρd m T +1). (24) We refer to (23) and (24) as the transformed m-model. Unlike the λ-model, it is difficult to characterize the exact optimal policy because the default penalty cost function M t is a general convex function (instead of a two-piece linear function in the λ-model). Thus, the optimal policy, if it exists, would be a general state-dependent policy. Below we provide simple heuristics and cost lower bounds based on linear approximations. 5.1 Linear Approximation We propose two types of piecewise linear functions to approximate the convex function M. As we shall see, each of the linear functions will lead to a lower bound and a heuristic for the m-model. Here and in the sequel, we suppress the time subscript without confusion. Two-piece linear approximation The first piece-wise linear approximation is generated by replacing the random variable D m t the mean value µ m in the M t function. More specifically, define with M (u) = p(u ρµ m ) +. (25) From Jensen s inequality, it is clear that M (u) M(u) for all u. Moreover, both functions have the same asymptotic slope p; see Figure 3(a). Three-piece linear approximation The above two-piece linear approximation only characterizes the first moment of random variable D m. Here, we further develop an approximation based on a three-piece linear function. This approximation, while more complicated to generate, takes into account the demand variability. We demonstrate how to construct the three-piece approximation. We generate a linear function Γ by constructing a tangent line to the convex curve M at point (ρµ m, M(ρµ m )); see Figure 3(a). 19

20 Let p be the slope of Γ. It can be shown that p = pf m (µ m ) and M(ρµ m ) = pρ ˆF m (µ m ). Thus, Γ(u) = p(u ρµ m ) + M(ρµ m ) = pf m (µ m )(u ρµ m ) + pρ ˆF m (µ m ). We term p the expected default penalty cost rate, which is the marginal cost rate when u = ρµ m. Let (ρa, 0) be the intersection point of Γ and the u-axis, and (ρa, M (ρa )) the intersection point of Γ and M. Now, define the three-piece linear function M(u) = max { M (u), Γ(u) } 0, if u ρa = Γ(u), if ρa < u ρa M (u), if ρa < u where A and A can be shown as, (26) A = µ m ˆF m (µ m ) F m (µ m ), A = µ m + ˆF m (µ m ) F m (µ m ). (27) To see this linear approximation takes into account the variability of the aggregated demand D m, notice that a = ρµ m ρa and a = ρa ρµ m, i.e., a = ρ ˆF m (µ m ) F m (µ m ), a = ρ ˆF m (µ m ) F m (µ m ). (28) Furthermore, for most unimodal distribution functions 7, we can show that ˆF m (µ m ) = (σ m ) 2 f(µ m ). (29) Thus, when the aggregated demand is more variable, a and a will be bigger (or equivalently, A will be smaller and A will be bigger). Figure 3(a) depicts A, A, a, a, two-piece linear function M (solid line), and three-piece linear function M (dashed line). The following proposition formally establishes the lower bound systems. The proof is straightforward and thus omitted. Lemma 2. By replacing M t with Mt (or M t ) in (23), the optimal cost of the resulting model forms a lower bound to the optimal cost of the transformed m-model. We refer to the resulting model with M t ( M t ) as the two-piece (three-piece) lower bound. 5.2 Lower Bound Solutions To find the optimal solution for the lower bound systems, we first define w = w + ρe D m(re m ) = w + ρµ m. 7 The demand functions tested include, but are not limited to, Poisson, Geometric, Negative-Binomial, Exponential, Gamma, and Normal, etc. 20