Collaborative product development: Managing supplier incentives for key component testing

Collaborative Product Development: Managing Supplier Incentives for Key Component Testing

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Collaborative Product Development: Managing Supplier Incentives for Key Component Testing Timofey Shalpegin PII: DOI: Reference:

S0377-2217(20)30121-1 https://doi.org/10.1016/j.ejor.2020.02.003 EOR 16323

To appear in:

European Journal of Operational Research

Received date: Accepted date:

4 July 2019 3 February 2020

Please cite this article as: Timofey Shalpegin, Collaborative Product Development: Managing Supplier Incentives for Key Component Testing, European Journal of Operational Research (2020), doi: https://doi.org/10.1016/j.ejor.2020.02.003

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Highlights • A high reward may prompt a supplier to release a low-quality component • A higher reward for component success may not lead to more testing by the supplier • A penalty for component failure alone is not sufficient to coordinate supply chain • The efficient contract implies a penalty for non-release of the component

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Collaborative Product Development: Managing Supplier Incentives for Key Component Testing Timofey Shalpegin Department of Information Systems and Operations Management University of Auckland Business School 12 Grafton Rd, 1010 Auckland, New Zealand [email protected] Abstract Supplier involvement in new product development has become increasingly common. When developing an innovative component for a new product, it may be sufficient for the component supplier to merely follow the buyer’s specifications and release the component without a good understanding of how well it will fit the final product. While this approach meets reasonable expectations of supplier involvement, it may increase the probability of the final product’s failure. The alternative for the supplier is to exceed expectations by learning more about the final product, performing extra tests, and gathering more information about component fit. This paper studies supplier incentives to exceed expectations before releasing the component for mass production. We develop a sequential non-cooperative game with endogenous information asymmetry. We then solve the game for reward-only and residual claimant contracts and compare the outcomes against the first-best outcome. We find that offering a higher reward to the supplier toward component success does not always lead to greater effort on the supplier side. Instead, this approach may backfire on the buyer by making the supplier less likely to test the component and, in some cases, more likely to release a low-quality component. We also find that introducing a penalty for component failure does not lead to coordination of the supply chain either. However, an efficient contract exists, but it requires the buyer to introduce a penalty for non-release of the component.

Keywords: supply chain management; component quality; product development; supplier involvement; game theory

1

Introduction

Suppose you need to do grocery shopping for a family dinner, but the shopping list has been prepared by another member of the household. You can either mechanically buy everything included in the list or carefully study it, guess what recipe the author of the list has in mind, identify potentially missing ingredients, call to confirm if there is enough of them at home, and so on. After returning home, you might jointly start cooking the dinner, but if an ingredient happens to be missing or in insufficient quantity—despite your strict adhesion to the shopping list—in the best 2

case, you will need to take another trip to the supermarket, and in the worst case—forget about the dinner. So, what is the right balance between checking the shopping list several times and risking failure at the cooking stage? The example above illustrates a situation where one could merely follow the instructions to get the job done, or, alternatively, go the extra mile to understand the instructions, and perhaps do a better job in the end. This paper extends this problem to the business environment where a buyer of a new product involves an external company to develop a key component for that product. If the development is successful, the company involved becomes the supplier of the developed component at the mass production stage. Next, we provide a few high-profile examples of such supplier involvement in product development. In 2003, the Boeing Company started on the development of a new aircraft, the Dreamliner. A key feature of the new aircraft was intended to be its considerably lower weight in comparison to the existing aircraft models. Pursuing this goal, Boeing decided to replace the traditional, heavy nickelcadmium battery used in the aviation industry with the novel and substantially lighter lithium-ion battery (Adolph, 2013). In 2005, Boeing approved Yuasa, a Japanese subcontractor, to develop and subsequently supply the lithium-ion batteries. Yuasa successfully developed the required battery to the specifications provided by Boeing, and by 2013 mass production of the lithium-ion batteries had started. However, shortly after Boeing made the Dreamliner available for commercial sale, several of the new aircraft caught fire, and Boeing had to ground the entire Dreamliner fleet for several months while they identified and fixed the cause of the problem. Although the grounding-related expenses were largely covered by insurance (Tsikoudakis, 2013), Boeing suffered a six-billion dollar cash drag during 2013 (Lowy and Freed, 2013), as well as losing sales and incurring damage to their reputation. Following the fire incidents, the cause of ignition and smoke was identified as the lithium-ion batteries overheating because the supplier had not performed additional tests to verify that their batteries would be compatible with aircraft systems. The supplier’s quality control was adequate for the requirements of the automotive industry but proved to be not rigorous enough for aviation (Wald and Mouawad, 2013). Commenting after the investigation, Yuasa’s President noted the following lessons for the future: “Instead of merely following instructions and making batteries, we should also study their instructions, collect data ourselves and make suggestions” (Kubota and Osada, 2013). In other words, the president was acknowledging they should have considered a range of costly activities to establish the fit of their component prior to releasing it to the buyer. As a result, although the supplier suffered from some reputation loss affecting their non-aviation business (Motavalli, 2013), no real penalties were applied for the battery failures (Cooper and Mukai, 2013). Another example involves the failed ExoMars 2016 mission by the European Space Agency. The Mars lander experienced problems at the descent stage and crashed into the surface of Mars. One of the causes of the accident was identified as a “mishap of management of subcontractors and hardware acceptance” (Tolker-Nielsen, 2017). The problem was that the hardware for a key component was developed by a supplier, whereas the main contractor developed the software. As

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a result, they were never properly tested together before the launch. The report on the accident concluded that the supplier should have been more involved in the software development, or at least in its validation. Both these examples follow the same pattern: Key components developed by external suppliers fail, but there appears to be no fault attributable to the suppliers as they provided the components according to the buyers’ specifications. The issue seems to be that the suppliers chose not to go the extra mile to test the components beyond what was specified by the buyers. As a result, the suppliers did not establish how well their component would fit the final product. An alternative explanation is that the suppliers knew the potential risks but decided to deliver the component regardless. In support of this explanation, Boeing’s supplier of battery chargers had a fire in their factory while testing their charger with Yuasa’s batteries, an event Yuasa was definitely aware of (Brewin, 2013). In either case, the question is-are such supplier decisions ex-ante beneficial for the supply chain, or do they damage the supply chain’s value? If the suppliers from the above examples had taken the further step of acquiring the extra information needed to establish component quality or fit, they could have made a well-informed decision on whether to risk releasing the component, or not release it at all. A development failure is a possible outcome of the component development process and in some cases, the buyer may even cover the supplier’s development costs irrespective of the outcome (Bhaskaran and Krishnan, 2009). Acknowledging failure at the development stage is often better for both parties than a final product failure after integration of the component in the final product. The objective of this research is to identify if suppliers have sufficient incentive to collect extra data, study general buyer specifications, and perform additional tests following the component’s development. As the buyer may not have enough expertise to perform and even identify the necessary tests or other activities, ensuring that the suppliers have taken care of this aspect is crucial. In this paper, we investigate the incentives different contracts create for suppliers, and the ways buyers can modify them to induce a better supply chain performance. In particular, we consider reward-only and residual claimant contracts. The reward-only contract does not impose any penalty on the supplier for component failure, whereas the residual claimant contract implies the buyer’s payoff is independent of the component’s success or failure, and the supplier bears all the associated risks. We develop a sequential game-theoretic model in which the supplier may choose to learn private information about the component quality before releasing the component. We find that providing a high reward to the supplier for the component success is not a panacea. Paradoxically, this can backfire on the buyer, causing the supplier to fail to perform enough tests or even release a lowquality component. We further discover that it is possible to design an efficient contract, but this contract requires the buyer to set a penalty for non-release of component by the supplier, contrary to the expectation that such a penalty could drive the supplier to release a low-quality or untested component. The remainder of the paper is organized as follows. First, Section 2 reviews the relevant liter-

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ature. In Section 3, we introduce an analytical model describing the collaborative product development process with the possibility of failure. Section 4 analyzes the first-best scenario, i.e., the case of a single decision-maker maximizing the total supply chain profit. Sections 5 and 6 deal with reward and residual claimant contracts, respectively. Further, Section 7 derives the efficient contract, i.e., the contract leading to the same supply chain profit as the first-best scenario. Section 8 relaxes several modeling assumptions. Finally, Section 9 provides conclusions and discusses our study’s applications and limitations.

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Literature Review

Our research relates to several streams of literature including product recalls, collaborative new product development, and economic literature on principal-agent models with endogenous information structure. We begin by reviewing the relevant product recall literature. Jarrell and Peltzman (1985) were among the first to analyze the total cost of product recalls in various industries. Their main finding is that the main cost incurred by a product recall is often the loss of goodwill. Moreover, a product recall produces significant negative externalities, implying losses for other stakeholders who are not responsible for the recall. Although debatable (Hoffer et al., 1988), this idea of goodwill loss spreading to the shareholders is widely supported in the literature (e.g., Rupp, 2004; Tse and Tan, 2011; Piramuthu et al., 2013; Filbeck et al., 2016). In line with this research, in our model, we allow for the respective goodwill losses of the supplier and the buyer, even though there is no direct fault attributable to the supplier in the cases we consider. There is an extensive literature on optimal quality inspection mechanisms and appraisal tests that can be implemented by the buyer (e.g., Baiman et al., 2000; Balachandran and Radhakrishnan, 2005; Babich and Tang, 2012; Rui and Lai, 2015; Dong et al., 2016). However, our setting is different as we consider a scenario where such inspections and tests are unfeasible due to a lack of expertise on the buyer’s side. Chao et al. (2009) analyze supplier incentives to improve product quality under different contracts outlining the rules for sharing the recall costs. Along similar lines, Mai et al. (2017) show that the provision of extended warranties can coordinate product quality. These models address cases when supplier fault can be proved, while our research deals with the scenarios where the supplier is not at fault as they have followed all the buyer’s instructions. Furthermore, they assume the component’s quality is known to the supplier and accordingly, they need to decide on the quality improvement efforts. However, this stage can only be reached after the supplier has established the component’s quality, whereas the focus of our paper is precisely modeling the supplier’s decision to learn about the component’s quality. While related research on supply chain disruption, including Yang and Babich (2014), a review by Snyder et al. (2016), Li (2017), and others, primarily focuses on the supply network design and assumes suppliers know the probability of their failure, this assumption does not hold in our setting. The literature on collaborative new product development has long recognized the importance of creating the right incentives within the supply chain to ensure the quality and cost of the final

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product. In particular, Kim (2000) studied how the buyer can subsidize supplier innovation to reduce component cost. Further, Bhaskaran and Krishnan (2009) investigated the buyer’s innovation and investment sharing with the supplier, while Kim and Netessine (2013) compared different procurement schemes leading to component cost reduction. More recently, Lambertini (2018) has investigated the effect of research and development efforts in the supply chain on product quality. Other mechanisms to improve the supplier’s efforts at the product development stage have been investigated by Bhattacharya et al. (2013) and Lee and Li (2018). Supplier development in the context of new product development has also been the subject of increased research attention (Modi and Mabert, 2007; Talluri et al., 2010; Lawson et al., 2015). Our research contributes to this literature stream in two ways: First, we model the supplier’s (rather than buyer’s) decision to learn about the component’s quality, by allowing the supplier to decide whether to learn private information and thus create information asymmetry. Second, we introduce the possibility of product failure adversely affecting both the supplier and the buyer through goodwill losses. Overall, our research attempts to bridge this literature stream with the product recall literature. From the agency theory perspective, our problem falls into a broad category of principal-agent models with endogenous information structures (Laffont and Martimort, 2002, p. 395). The main feature of such models is that the agent (the supplier) does not know its type (the component quality) ex-ante, and so needs to choose whether to learn it. In our model, after optional learning of its type, the agent needs to decide whether to release the component. Malcomson (2009), extending research by Lambert (1986), provides a general framework for the analysis of such principal-agent models that shows that the first-best solution cannot be achieved. Our problem introduces additional details specific to collaborative product development. Specifically, the supplier payoff in our model is directly dependent on the outcome of the project in terms of the goodwill loss rather than solely through transfers from the buyer. Furthermore, we extend the range of admissible contracts to derive an efficient contract, i.e., the contract leading to the first-best expected supply chain profit. A considerable body of principal-agent literature dealing with endogenous information focuses on deriving the optimal menu of contracts to screen the agent’s type after acquiring private information. Cr´emer and Khalil (1992) investigate a setting similar to ours, with the exception they do not allow the agent to fail and, consequently, the principal (the buyer) cannot suffer from the agent’s failure. They find that the optimal contract does not incentivize the agent to gather information (perform tests), a result that is not in agreement with our findings. In a subsequent study, Cr´emer et al. (1998) allow the agent to gather information prior to the contract offer by the principal, which makes the agent’s decision strategic. Szalay (2009) further develops this problem but allows the information to be continuous in the amount of effort exerted to gather it. In our model, however, the agent starts the information gathering after receiving the contract terms; otherwise, they would never start the component development in the first place. Lewis and Sappington (1993) consider the problem that the agent’s efforts may or may not lead to private information that is unobservable by the principal, and thus their focus is not the agent’s decision to gather information.

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Kr¨ ahmer and Strausz (2011) model the unobservable pre-project acquisition of cost information by the agent. However, their model implies an initial asymmetric information structure, and thus the analysis focuses on deriving the optimal menu of contracts. In contrast, in our setting, the initial information (before learning the component quality) is symmetric, and hence screening mechanisms are not applicable. Finally, Roger (2013) investigates a problem similar to ours but considers a contract that cannot be conditioned on the final outcome.

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Model

In our model, a buyer (“he”) involves a supplier (“she”) in the development of a component for a new product. In line with the literature on product recalls and collaborative product development (e.g., Chao et al., 2009; Kim and Netessine, 2013; Yoo et al., 2015), we assume both players are risk-neutral and their objective is to maximize their respective expected profits. The buyer offers a contract to the supplier, and the supplier decides (1) whether to test the component and learn its quality, and (2) whether to release the component for mass production. Below, we discuss the event sequence and our key assumptions in detail. At the first stage, the buyer offers the supplier a contract specifying the supplier’s reward w if the component works well after integration in the final product. The reward w represents the net value the supplier expects to receive from procuring the component at the mass production stage if the component works as expected. This is a typical assumption in the literature on collaborative product development (e.g., LaBahn and Krapfel, 2000; Shalpegin et al., 2018). The contract may also specify the penalty p ≥ 0 the supplier should pay if the component fails, which is also common

in the literature (e.g., Yang et al., 2009; Yoo et al., 2015). Figure 1 illustrates the sequence of events as well as the payoffs for the buyer and the supplier. In the figure, B stands for the buyer, S—for the supplier, and N refers to nature, i.e., to the realization of random variables. The black nodes are the terminal nodes of the game, and we report the players’ payoffs in brackets, starting with the buyer’s payoff. At the next stage, the supplier needs to decide whether to collect the necessary data, identify the scale and the scope of the necessary tests, and perform the tests (for simplicity, from now on we will refer to these activities as performing the tests, or testing the component) to learn the quality of the new component at a cost of t > 0. We assume that the buyer knows the probability distribution of t, denoted as G(t) and g(t) for cumulative distribution and probability density function, respectively, while the supplier knows the exact value of t. This assumption reflects the idea that the supplier has more knowledge about the component and possible tests than the buyer. We define the quality of a component as the likelihood that the final product does not fail, similarly to Chao et al. (2009). If the supplier chooses to perform the tests, she learns the component’s quality, i.e., the success probability θ, which is drawn from a probability distribution with the cumulative density function F (θ) and the support [0, 1]. If the quality is not learned, the supplier’s knowledge remains limited to the distribution F (θ). This modeling approach allows us to capture

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the difference in the degree of uncertainty that the supplier experiences before and after performing the tests. The buyer only knows the probability distribution of the quality and cannot gain any more precise knowledge. Likewise, the buyer does not know whether the supplier has performed tests or not, which is represented by the dotted rectangle in Figure 1.

Figure 1: Simplified Game Tree At the next stage, the supplier needs to decide whether to release the component for mass production. If the supplier releases the component and it proves successful (which happens with probability θ), the buyer will pay the supplier the reward w. We denote the component total added value to the buyer as r and the corresponding supplier production cost as c. To avoid trivial scenarios, we assume r − c ≥ 0. It is easy to see that the buyer will always set the reward so that w ∈ [c, r]. Otherwise, the supplier never releases the component (if w < c), or the buyer is better off by abandoning the project (if w > r).

If the component fails (with probability 1 − θ), both the buyer and the supplier incur losses

of lb > 0 and ls > 0, respectively. These losses incorporate goodwill losses, as well as losses due to project delays, product recalls, and compensation to the customers (if any). Furthermore, the supplier will need to pay a penalty of p to the buyer if the contract specifies such a penalty. The supplier may choose not to release the component. In this case, the collaboration terminates, and the supplier’s payoff will be 0 if she has not performed the tests, or −t if she has performed the tests.

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To differentiate the players’ payoffs at different stages of the game, we introduce the notion of history. In particular, there are two different stages after the buyer offers the contract. At Stage 1, the supplier has decided whether to test the component, which is formally captured as {T, nT },

i.e., the history at Stage 1 is T if the supplier has tested it, and nT otherwise. At Stage 2, the supplier has decided whether to release the component for mass production, and the history vector is ({T, nT }, {R, nR}), where R stands for the release and nR—for the non-release.

Overall, we have developed a non-cooperative sequential game between two players. We will

next rely on backward induction to find the equilibrium strategies based on the concept of subgame perfect Nash equilibrium.

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First-best Analysis

As the first step, we analyze the outcome arising in the integrated supply chain, i.e., in the presence of a single decision-maker. We refer to such an outcome as the first-best outcome, and we will use it as a benchmark for our further analysis.

4.1

Release Decision

Solving the problem backwards, we start with the decision whether to release the component. Suppose the integrated firm has already tested the component and observed its quality θ. Recall that for the integrated supply chain we sum the supplier’s and the buyer’s payoffs. Hence, the expected payoff from the component release is given as π ∗ (T, R) = θ(r − w + w − c − t) + (1 − θ)(−lb + p − ls − p − t) = θ(r − c) − (1 − θ)(ls + lb ) − t (1) We next explain Equation (1) in detail, as follows: With probability θ, the component works well, and the integrated firm receives the payoff r − c. Note that the reward w as well as the penalty

p cancel out as they are just transfer payments within the boundaries of the same firm in this case. With probability 1−θ, the component fails, and the firm incurs losses of ls +lb . Finally, irrespective of the outcome, the firm has already carried out the tests at the cost of t. If the component is not released, the payoff is π ∗ (T, nR) = −t, which reflects that the firm has

incurred the testing costs, and it will not receive any other gains or losses. Comparing the payoffs allows us to formulate the optimal release policy for the case when the component has been tested, which is captured by Proposition 1(a). Meanwhile, if the integrated firm has not tested the component, the payoffs will be different. In particular, if the firm chooses to release the component, the payoff is π ∗ (nT, R) = θ(r − c) + (1 −

θ)(−lb − ls ), where θ is the mean value of θ. In this case, the firm has to rely on the mean value of θ

as its exact value cannot be known without carrying out the tests. If the component is not released, the firm’s payoff is π ∗ (nT, nR) = 0. The optimal release policy is given by Proposition 1(b). 9

Proposition 1 (a) If the tests have been performed, the optimal action is to release the component if θ ≥ θ∗ , and not to release the component if θ < θ∗ , where θ∗ =

ls +lb r−c+ls +lb .

(b) If the tests have not been performed, the optimal action is to release the component if θ ≥ θ∗ ,

and not to release the component if θ < θ∗ .

We have included all proof in the appendix. The important insight is that for the informed integrated firm, there exists a certain welldefined threshold for θ, which allows the firm to decide whether to release the component, and the firm releases the component only if the observed component quality θ is above the threshold. This threshold, θ∗ , increases in the integrated firm’s losses and decreases in the firm’s gains. This relationship can be rewritten as θ∗ =

Loss Gain+Loss ,

where Loss stands for the total loss incurred in the

case of failure, ls + lb , and Gain is the total reward acquired in the case of success, r − c. As we

will see in the next sections, a similar critical ratio reoccurs under various admissible contracts.

4.2

Testing Decision

Going backwards along the decision tree, we can now derive the expected profit for the integrated firm given its optimal release decision. If the firm chooses to perform the tests, it is
π ∗ (T ) = 1 − F (θ∗ ) Eθ≥θ∗ {θ(r − c) − (1 − θ)(ls + lb )} − t,

(2)

where Eθ≥θ∗ is the conditional expectation taken with respect to θ so that θ ≥ θ∗ . Technically, π ∗ (T ) consists of several parts: (a) with probability 1 − F (θ∗ ) the component will have sufficient

quality to be released, and the firm’s payoff will depend on the component’s success or failure, (b) with probability F (θ∗ ) the component’s quality will be so low that the firm will not release it, and

hence, the payoff is 0, and (c) in either case, the testing cost t is spent. If the firm opts not to perform the tests, the expected profit is π ∗ (nT ) = max{π ∗ (nT, R), π ∗ (nT, nR)} = θ(r − c) − (1 − θ)(ls + lb )


+

,

(3)

where the notation (x)+ means max{x, 0}. In this case, the firm will release the component if the expected profit from this decision is non-negative, and will not release it otherwise. The expected profit cannot be negative because the firm will not release the component in that case. Clearly, the integrated firm performs the tests only if π ∗ (T ) ≥ π ∗ (nT ), in particular,

+ 1 − F (θ∗ ) Eθ≥θ∗ {θ(r − c) − (1 − θ)(ls + lb )} − t ≥ θ(r − c) − (1 − θ)(ls + lb )

(4)

Isolating t in the above inequality and simplifying it, we can capture the rule for when to test the component in the first-best scenario. Proposition 2 shows that the testing decision is of a threshold type, i.e., it is optimal to test if the testing cost is below the threshold, and it is optimal not to test if the testing cost is above the threshold.

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Proposition 2

(a) If π ∗ (nT, R) ≥ 0, the component testing is optimal if t≤

Z

0

θ∗


(1 − θ)(ls + lb ) − θ(r − c) f (θ) dθ ≡ ta ,

(5)

and otherwise it is optimal not to test. (b) If π ∗ (nT, R) < 0, component testing is optimal if t≤

Z

1

θ∗


θ(r − c) − (1 − θ)(ls + lb ) f (θ) dθ ≡ tb ,

(6)

and otherwise it is optimal not to test.

In the above proposition, we use the notation ≡ to define the variables. In particular, we define

the testing cost thresholds for cases (a) and (b) as ta and tb , respectively.

Proposition 2 establishes that the optimal testing rule is different depending on whether π ∗ (nT, R) ≥

0 or π ∗ (nT, R) < 0. As we will see in the subsequent sections, the same holds for the contracts

we consider between an independent buyer and supplier. Note that π ∗ (nT, R) ≥ 0 means the firm receives a positive expected profit if it releases an untested component. This scenario is likely to

occur when the potential loss is sufficiently small, or the potential gain far outweighs the loss. In that case, releasing the component “blindly”, i.e., without learning its quality, is better than not releasing it at all. Such an environment might prevail in some industries; for example, Thirumalai and Sinha (2011) provide empirical confirmation that this could be the case for medical device manufacturing. At the same time, π ∗ (nT, R) < 0 represents the opposite situation whereby the firm will never release an untested component, i.e., a “blind” component release is not optimal in this case. This could be relevant for industries with relatively high goodwill losses following a product failure, e.g., the toy industry (Freedman et al., 2012). Having established the benchmarks for component testing and release decisions, we will now turn to the analysis of some typical supply chain contracts. Our primary objective is to see whether these contracts are capable of incentivizing the supplier to follow the first-best decision.

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Reward Contract

We start with the case where the supplier cannot be obliged to pay any penalty in the case of component failure, i.e., p = 0, while she still bears the reputation loss ls > 0. Practitioners may deem this contract type as the most reasonable because the suppliers were not at fault in any of our motivational examples. Charging the supplier for component failure in these cases could be considered unjustified. Another possible rationale for this contract type can be that the supplier has limited liability, and hence, no significant penalty can be charged as the supplier might default. This is particularly relevant for relatively small suppliers engaged in large complex projects (Zsidisin et al., 2000).

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5.1

Release Decision

The first step is to find out when the supplier chooses to release the component. If she has tested the component, the expected profits for the cases of release and non-release are π ˆs (T, R) = θ(w−c)−(1−θ)ls −t and π ˆs (T, nR) = −t, respectively. If the supplier has not tested the component,

the expected profit is π ˆs (nT, R) = θ(w − c) − (1 − θ)ls if she releases it, and π ˆs (nT, nR) = 0 otherwise. Structurally, the expected profits are similar to those under the first-best scenario with two important exceptions: (a) the reward in case of success is now w ≤ r, and (b) the loss in case of failure is now ls < ls + lb .

The following proposition captures the supplier equilibrium strategy in this subgame. It is structured similarly to Proposition 1 to ensure an easy comparison. Proposition 3 (a) If the tests have been performed, the optimal action is to release the component ˆ and not to release the component if θ < θ, ˆ where θˆ = ls . if θ ≥ θ, w−c+ls ˆ (b) If the tests have not been performed, the optimal action is to release the component if θ ≥ θ,

ˆ and not to release the component if θ < θ.

Although featuring a similar structure, the release threshold is not the same as for the first-best case. An important difference is that the supplier does not take into account the buyer’s reputation loss, lb , when deciding on the component release. Comparing the release threshold for the first-best scenario and the reward contract, we formulate the following corollary. Corollary 1 It holds that (a) θ∗ = θˆ if w = w<

ls r+lb c ls +lb .

ls r+lb c ls +lb ,

(b) θ∗ > θˆ if w >

ls r+lb c ls +lb ,

and (c) θ∗ < θˆ if

Depending on the value of w, the supplier may be more (or less) incentivized to release the component relative to the first-best scenario. Furthermore, the buyer can set a certain value of w, such that the supplier release threshold coincides with the first-best release threshold. Figure 2 illustrates how the release threshold depends on the reward w and compares it to the first-best release threshold. The finding that the supplier is more likely to release the component as the reward w is increasing is significant. If w is relatively low, the supplier does not release the components that would be released in the first-best scenario. However, if w increases beyond the point of

ls r+lb c ls +lb ,

the

supplier will release unreliable low-quality components that would not be released in the first-best scenario (represented by the gray region in Figure 2). With increasing w, the potential gain for the supplier increases while the loss remains unchanged, which then prompts her to release low-quality components. This finding echoes the well-known double-marginalization effect in supply chains, which leads to inefficient ordering decisions due to inappropriate risk allocation among the players (e.g., Li et al., 2013). In our context, this effect extends to the component release decision. We further proceed with the analysis of the supplier testing decision given the equilibrium release decision.

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Parameters: θ ∼ U (0, 1), c = 2, lb = 2, ls = 3, and r = 10 Figure 2: Quality Thresholds for Component Release

5.2

Testing Decision

The next step is to derive the condition defining whether the supplier chooses to test the component in the first place. The supplier will perform the tests as long as her expected payoff after testing is not less than the expected payoff without testing, π ˆs (T ) ≥ π ˆs (nT ),

(7)


ˆ E ˆ{θ(w − c) − (1 − θ)ls } − t and where π ˆs (T ) = 1 − F (θ) θ≥θ


+ π ˆs (nT ) = max{ˆ πs (nT, R), π ˆs (nT, nR)} = θ(w − c) − (1 − θ)ls . The immediate observation is that limls →0 θˆ = 0, in which case inequality (7) does not hold,

which means that in the absence of the loss ls , the supplier will never test the component prior to its release. This result is to be expected, as if there is no loss in the case of failure, then the supplier will always try the new component, which means that there is no rationale for the supplier to test the component before releasing. Therefore, the only incentive to perform the component tests arises when loss is incurred in the case of failure. The following proposition provides the exact rule for how the supplier will decide whether to test the component. Proposition 4

(a) If π ˆs (nT, R) ≥ 0, the component testing is optimal if t≤

Z

0

θˆ


(1 − θ)ls − θ(w − c) f (θ) dθ ≡ tˆa ,

and otherwise it is optimal not to test.

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(8)

(b) If π ˆs (nT, R) < 0, the component testing is optimal if t≤

Z

1

θˆ


θ(w − c) − (1 − θ)ls f (θ) dθ ≡ tˆb ,

(9)

and otherwise it is optimal not to test. Interestingly, tˆa is decreasing in w, which is not intuitive. This means that the more the supplier receives in the case of the component’s success, the less likely she is to test the component. Figure 3 illustrates this phenomenon, as follows: The testing region under the reward contract shrinks for high w as w increases further. This relationship for higher-reward-but-less-testing might be puzzling at first glance, and yet, there is a solid reason for it if we link it to the supplier’s release decision. As the supplier’s reward increases, the supplier’s expected profit from the “blind” component release (i.e., the release of an untested component) keeps increasing while the value of the information provided by the component testing increases at a slower pace. In other words, if the reward is very high, the supplier will try to achieve it by releasing even a low-quality component (recall Figure 2). Hence, the value gained from having the component tested becomes relatively lower as, most likely, the test will show the supplier needs to release the component. If the result is so predictable, why then would she test in the first place? This is why testing becomes less attractive with higher reward values. Note that for low reward values, the opposite holds. As shown in Figure 3, a higher reward leads to a larger testing region, i.e., tˆb is increasing in w. The reason for this is that the supplier will never release an untested component in this case, as the supplier needs to choose between testing the component and leaving the project (by not releasing the component). As w increases, the benefit of taking the former option increases while the benefit of the latter remains unchanged. Hence, the supplier finds it optimal to test the component at a larger testing cost as w increases. Proposition 5 sheds light on the possibility of achieving the efficient outcome under the reward contract. Proposition 5 If w is such that θˆ = θ∗ , then the testing cost threshold under the reward contract is always less than the testing cost threshold for the first-best case. The above result means the first-best outcome cannot be achieved under the reward contract. If the buyer sets the payment so that the supplier releases the component according to the first-best release rule, the supplier will more likely choose not to test the component relative to the first-best case. Having established the equilibrium supplier strategy, the next step is to analyze the optimal buyer’s action, which is choosing the reward w. One can find the optimal w from the buyer’s

14

Parameters: θ ∼ U (0, 1), c = 2, lb = 2, ls = 3, and r = 10 Figure 3: Testing Cost Thresholds for Component Testing objective function, given as follows:

π ˆm


 ˆ E ˆ{θ(r − w) − (1 − θ)lm }  G(tˆa ) 1 − F (θ)  θ≥θ 

= + 1 − G(tˆa ) θ(r − w) − (1 − θ)lm , if π ˆs (nT, R) ≥ 0  
 G(tˆ ) 1 − F (θ) ˆ E ˆ{θ(r − w) − (1 − θ)lm }, if π ˆs (nT, R) < 0 b θ≥θ  R θ(r − w) − (1 − θ)l + G(tˆ ) θˆ (1 − θ)l − θ(r − w)
f (θ) dθ, if w ≥ a 0 b b = G(tˆ ) R 1 θ(r − w) − (1 − θ)l
f (θ) dθ, if w < b b θˆ

1−θ l θ s 1−θ l θ s

+c

(10)

+c

We now further explain Equation (10). If the supplier receives a non-negative expected profit from the release of an untested component (ˆ πs (nT, R) ≥ 0), with probability G(tˆa ), she will find that
ˆ , the testing costs are sufficiently low to proceed with the tests and then with probability 1 − F (θ) she will release the component. In such a case, the buyer gets his expected payoff given that the ˆ However, with probability 1 − G(tˆa ), component quality is above the supplier’s release threshold θ.

the testing cost will be too high, and the supplier will release the untested component. In this case, the buyer’s payoff is based on the mean value of the component’s quality. The case of π ˆs (nT, R) < 0

is similar, with the exception that the supplier will not release the untested component. Accordingly, we have simplified the expressions, and isolated the conditions for w. The exact functional form of the optimal w depends on the probability distributions of the component quality and testing costs.

15

6

Residual Claimant Contract

The literature suggests that for endogenous information agency problems, the contracts that make the supplier the residual claimant for the total profit can achieve the first-best outcome (Laffont and Martimort, 2002). The reasoning is straightforward: If the supplier needs to maximize the same objective function as the buyer, but then gives away some fixed amount, her optimal decisions will be identical to the buyer’s. Arya et al. (2015) show that making the retailer a residual claimant helps coordinate a multi-tier supply chain. We would like to investigate whether this logic applies to our specific problem. The application of the residual claimant contract in practice remains rather limited because it implies full transparency with regard to the overall supply chain profit. Nevertheless, evidence of using the residual claimant contract in real life exists in various supply chain settings featuring moral hazard. For example, Steiner (2011) illustrates its application in the wine industry, and Kim et al. (2007) show that using the fixed-price contract in after-sales service supply chains effectively makes the supplier the residual claimant. The idea of the residual claimant contract is that the buyer should receive a fixed amount K irrespective of the outcome. To ensure that the buyer gets K in case of success, the reward should be set as w = r − K; in this way, the buyer receives r − w = r − (r − K) = K. To ensure he gets K in the case of failure, the penalty should be p = lb + K, so that the buyer receives −lb + p = −lb + (lb + K) = K. Under this payment structure, the supplier receives r − c − K

and −ls − lb − K in the case of success and failure, respectively. Hence, the supplier becomes the

residual claimant receiving the supply chain gains or losses (depending on the outcome), less the constant value K.

6.1

Release Decision

Similarly to the previous sections, we start with the analysis of the release decision. If the supplier has tested the component, her expected profits for the cases of release and non-release are π ˜s (T, R) = θ(r−c)−(1−θ)(ls +lb )−K −t and π ˜s (T, nR) = −t, respectively. If she has not tested the component,

the expected profits are π ˆs (nT, R) = θ(r − c) − (1 − θ)(ls + lb ) − K and π ˜s (nT, nR) = 0.

The following proposition formalizes the supplier’s optimal rule for when to release the compo-

nent. Proposition 6 (a) If the tests have been performed, the optimal action is to release the component ˜ and not to release the component if θ < θ, ˜ where θ˜ = ls +lb +K . if θ ≥ θ, r−c+ls +lb

˜ (b) If the tests have not been performed, the optimal action is to release the component if θ ≥ θ, ˜ and not to release the component if θ < θ. We can immediately see that θ˜ > θ∗ , which means the supplier will not release the component unless it is of the exceptionally high quality. This results in a lower efficiency for the supply chain as the components that should be released will be withheld due to the supplier being excessively 16

cautious due to the failure penalty. Figure 2 illustrates this finding: The component release region is smaller than for the first-best scenario. Similarly to the reward contract, a higher reward does decrease the release threshold. However, unlike the reward contract, the supplier will never release low-quality components that would not be released in the first-best scenario.

6.2

Testing Decision

We turn to the analysis of the supplier’s decision whether to test the component. The supplier chooses to test the component if π ˜s (T ) ≥ π ˜s (nT ),




˜ E ˜{θ(r − c) − (1 − θ)(ls + lb ) − K} − t and where π ˜s (T ) = 1 − F (θ) θ≥θ

π ˜s (nT ) = max{˜ πs (nT, R), π ˜s (nT, nR)} = θ(r − c) − (1 − θ)(ls + lb ) − K

(11)
+

.

We can see that in general, the testing region under (11) does not coincide with the region under

(4), which may indicate the contract does not lead the supply chain to the first-best outcome. Next, to illustrate it more clearly, we explicitly show the testing cost thresholds. Proposition 7

(a) If π ˜s (nT, R) ≥ 0, the component testing is optimal if ˜ − t ≤ ta + F (θ)K

Z

θ˜


θ(r − c) − (1 − θ)(ls + lb ) f (θ) dθ ≡ t˜a

θ∗

(12)

and otherwise it is optimal not to test. (b) If π ˜s (nT, R) < 0, the component testing is optimal if
˜ K+ t ≤ tb − 1 − F (θ)

Z

θ˜

θ∗


(1 − θ)(ls + lb ) − θ(r − c) f (θ) dθ ≡ t˜b

(13)

and otherwise it is optimal not to test. We can see that the testing thresholds t˜a and t˜b do not generally coincide with those for the first-best scenario, ta and tb , respectively. From (12)-(13) and as illustrated by Figure 3, it is clear that for some parameters, the supplier is even more likely to test the component than the integrated firm, i.e., t˜a > ta and t˜b > tb . This excessive testing occurs if θ˜ is sufficiently high for the case of π ˜s (nT, R) < 0 and sufficiently low for the case of π ˜s (nT, R) ≥ 0 as represented by the gray region in Figure 3. However, contrary to common intuition, the excessive testing does not constitute an efficient (or better than efficient) outcome. The issue is that the supplier performs the tests even if the testing costs are prohibitively high. At the same time, the supplier is much less likely to release the component to the buyer due to the potential for high losses consisting of both the goodwill loss and the contract penalty. This creates a curious situation where the supplier is likely to test the component, but unlikely to release it afterward. Similarly to the reward contract, if the supplier’s reward increases to the point that her expected profit from releasing an untested component turns positive, any further increase in the reward value 17

leads to a decreasing testing probability, i.e., t˜a is decreasing in w. The reasoning is the same as for the reward contract: The expected supplier profit from releasing the untested component increases in w faster than the value obtained from the component testing, which makes the latter less attractive. The underlying reason why the residual claimant contract does not achieve an efficient outcome is that the supplier has an outside option in our model, i.e., she can choose not to release the component and thus avoid the risk of paying the penalty to the buyer. This option creates sufficient distortion in the model to prevent the residual claimant contract from achieving the efficient (firstbest) outcome.

7

Efficient Contract

As can be seen, the residual claimant contract does not achieve the first-best outcome because it provides a free exit option for the supplier. In this section, we extend our original model so that it becomes possible to construct a contract that achieves the first-best outcome, and discuss its potential practical implications. We call this contract an efficient contract. The key to the construction of such a contract is that we need to extend the supplier’s liability to the exit option, i.e., to cover the case where the supplier decides not to release the component. In other words, the supplier needs to be compelled to pay the penalty even if she does not release the component. This idea is rather counter-intuitive. Common sense would encourage buyers to make the exit option as easy as possible for the supplier, so that the supplier will have no incentive to release a low-quality component. Some buyers may even cover the supplier’s cost for the component development, irrespective of whether it is released (Bhaskaran and Krishnan, 2009). Yet, as we have already seen, this logic can backfire on the buyer. Thus, we proceed with constructing an efficient contract incorporating a penalty for non-release of the component by the supplier. Let us denote the penalty for non-releasing the component as p0 > 0. Note that this penalty is not captured by the initial game tree as presented in Figure 1. Proposition 8 If w = r − K, p = lb + K, and p0 = K, where K < r, the first-best outcome is

achieved when the following conditions are satisfied: π ∗ (T ) ≥ π ∗ (nT ) and K ≤ π ∗ (T ).

The conditions of the proposition are not particularly restrictive. They tell us that this contract leads to an efficient outcome when it is optimal to test the component from the integrated firm’s perspective, π ∗ (T ) ≥ π ∗ (nT ), and if the buyer sets a figure K less than the expected total profit

if the component is tested, K ≤ π ∗ (T ). If from the integrated firm’s perspective it is not optimal

to test the component, we end up in with a trivial case, where the only decision is whether to release the component, and the first-best outcome can be achieved much more easily using simpler contracts. As for the value of K, it is easy to show that it is not in the buyer’s best interest to set K > π ∗ (T ) as it makes the supplier likely not to test or release the component, thus leaving the buyer with a payoff of zero. 18

The efficient outcome is achieved through the elimination of the free exit option, i.e., the option of not releasing the component after the supplier has learned its quality. The buyer should set a penalty of p0 = K for non-release of the component. In such a case, the supplier’s incentives are perfectly aligned with the buyer’s. Irrespective of the supplier decision on the component release and the outcome if the component is released, the buyer receives a fixed value K, leaving the supply chain profits (or losses) to the supplier. Effectively, it makes K a sunk cost for the supplier who now maximizes the objective coinciding with the first-best objective. From a practical perspective, our finding means that managers should try to eliminate the free exit option for their suppliers if the decision phase is preceded by action toward the possible acquisition of private information. Such an approach is not always intuitive as it suggests penalizing the agent for playing safe, which the principal may mistakenly perceive as a desired course of action.

8

Model Extensions

8.1

Development Efforts

In addition to choosing to test the component, the supplier may choose to exert effort to improve the component quality. In this extension, we study the effect of the supplier’s quality improvement efforts on the efficient contract. In particular, we assume that if the supplier chooses to test the component and observes θ, she may further exert effort at a cost ce > 0 and improve the success probability of the component to θH such that θH > θ. We assume ce ≤ θH (r−c)−(1−θH )(ls +lb )−t

to avoid the trivial case in which exerting effort is never optimal. We extend our notion of history by adding the parameters {E, nE}, where E means the supplier chooses to exert effort to improve

the component quality, and nE means the supplier chooses not to do so and proceeds directly to the decision on whether to release the component. 8.1.1

Release Decision

We start our analysis by considering the first-best scenario. The component release decision comes after the effort decision. For this reason, the decision structure remains the same as described by Proposition 1 if the integrated firm has not exerted improvement effort. If the firm has exerted effort, the decision structure will be similar, with the only difference being that the firm needs to compare θH to the threshold θ∗ to determine whether to release the component. This logic holds for both the integrated firm and the efficient contract constructed in Section 7. Hence, the efficient contract still ensures the first-best release threshold θ∗ . However, we also need to investigate the effort and testing decision. For our analysis, we will only consider the case of θ < θ∗ < θH , as otherwise, there is no benefit in exerting effort and the analysis is identical to the base model analysis. In other words, we analyze the case in which it is not optimal to release the original component, but it is optimal to release the improved component.

19

8.1.2

Effort Decision

If the component has not been tested, its quality is unknown, and thus it cannot be improved. Hence, all the expected profit functions for this case are identical to those in the base model. If the component has been tested, the firm will release the component and the first-best expected profit from exerting effort is as follows: πe∗ (T, E) = πe∗ (T, E, R) = θH (r − c) − (1 − θH )(ls + lb ) − t − ce .

If the firm chooses not to exert effort, it will not release the component and its expected profit

will be the same as in the base model: πe∗ (T, nE) = πe∗ (T, nE, nR) = π ∗ (T, nR) = −t. The integrated firm will exert effort if πe∗ (T, E) ≥ πe∗ (T, nE).

Now consider two separate firms and the contract proposed in Section 7 with w = r − K,

p = lb + K, and p0 = K, where K < r. The supplier’s expected profit if exerting effort is

πse (T, E) = θH (w −c)−(1−θ)(p+ls )−t−ce = θ(r −c)−(1−θ)(ls +lb )−t−ce −K = πe∗ (T, E)−K. If the supplier chooses not to exert effort, she will not release the component, and the expected

profit will be the same as in the base model, i.e., πse (T, nE) = πse (T, nE, nR) = πs (T, nR) = −t − p0 = −t − K = πe∗ (T, nR). The supplier will exert effort if πe∗ (T, E) − K ≥ πe∗ (T, nE) − K, which is equivalent to πe∗ (T, E) ≥ πe∗ (T, nE). Therefore, the proposed contract achieves the first-

best effort choice in equilibrium. 8.1.3

Testing Decision

In the first-best scenario, if the integrated firm chooses to test the component, its expected profit

is πe∗ (T ) = π ∗ (T ) + F (θ∗ ) πe∗ (T, E, R) + t = 1 − F (θ∗ ) Eθ≥θ∗ {θ(r − c) − (1 − θ)(ls + lb )} +
F (θ∗ ) θH (r − c) − (1 − θH )(ls + lb ) − ce − t. In other words, the expected profit is equal to

the expected profit under the baseline scenario, plus the extra profit the firm will make from the improvement option if the initial component quality proves to be too low, and the testing cost is added to avoid double-counting. If the firm does not perform the tests, no improvement is possible, and hence, πe∗ (nT ) =
+ π ∗ (nT ) = θ(r − c) − (1 − θ)(ls + lb ) . The firm chooses testing if πe∗ (T ) ≥ πe∗ (nT ).

Now consider the supplier’s expected profit if she chooses to test under the contract from
Section 7, as follows: πse (T ) = 1 − F (θ∗ ) Eθ≥θ∗ {θ(w − c) − (1 − θ)(p + ls )} + F (θ∗ ) θH (w − c) −

(1 − θH )(p + ls ) − ce − t = 1 − F (θ∗ ) Eθ≥θ∗ {θ(r − c) − (1 − θ)(lb + ls )} + F (θ∗ ) θH (r − c) − (1 −
θH )(lb + ls ) − ce − t − K = πe∗ (T ) − K. If the supplier does not perform the tests, her expected profit is πse (nT ) = θ(w−c)−(1−θ)(p+
+
+ ls ) = θ(r − c) − (1 − θ)(ls + lb ) − K = πe∗ (nT ) − K. The supplier chooses to test the component

if πe∗ (T ) − K ≥ πe∗ (nT ) − K, which is equivalent to πe∗ (T ) ≥ πe∗ (nT ). Accordingly, the supplier’s

testing decision is equivalent to the first-best testing decision. This means the efficient contract achieves the first-best outcome even if the supplier exerts effort and improves the component quality.

20

8.2

Multiple Suppliers

It is also possible that the final product contains multiple key components developed by different suppliers. In this extension, we consider that two suppliers are developing key components and will only receive the reward if both components work well once integrated into the final product. We show that the modification of the contract suggested in Section 7 will achieve the first-best outcome. In particular, we consider the contract with wi = r − cj − Ki , pi = lb + lsj + Ki , and p0i = Ki , where i, j ∈ {1, 2}, i 6= j, and K1 + K2 ≤ r. Adding the indices i and j to the variables indicates that a particular variable is attributed to suppliers i and j, respectively.

We assume the suppliers know each other’s probability distribution for component quality Fi (θ), where i ∈ {1, 2}. If the suppliers choose to test the component, they will learn their component’s qualities θi . The probability of the final product’s success is then θ1 θ2 ; we, therefore, denote the

cumulative distribution function of the joint probability distribution of θ1 θ2 as F12 (θ). For this extension, we assume information sharing between suppliers, i.e., the test results from one supplier are shared with the other. If this assumption does not hold, information asymmetry will emerge between suppliers, which could lead to no contract achieving the first-best outcome. This scenario represents a promising direction for future research. 8.2.1

Release Decision

As before, we start our analysis by considering the first-best scenario. If at least one component is not released, the overall project fails, whereas if both are released, the expected profit (if both ∗ (T , T , R , R ) = θ θ (r − c − c ) − (1 − θ θ )(l + l + l ) − 2t. components have been tested) is πm 1 2 1 2 1 2 1 2 1 2 s1 s2 b ∗ (T , T , nR , nR ) = π ∗ (T , T , nR , R ) = −2t. The profit from non-release of the components is πm 1 2 1 2 i j m 1 2

∗ (T , T , R , R ) ≥ π ∗ (T , T , nR , nR ), Hence, the integrated firm will release both components if πm 1 2 1 2 1 2 m 1 2

which is equivalent to θ1 θ2 ≥ θ∗ , where θ∗ is defined in Section 4. If the firm has tested only com-

ponent i, the release condition will be θi θj ≥ θ∗ , where i, j ∈ {1, 2} and i 6= j. If the firm has not

tested any of the components, they will be released if θ1 θ2 ≥ θ∗ .

Now consider two independent suppliers, each offered a contract with wi = r − cj − Ki , pi =

lb + lsj + Ki , and p0i = Ki , where i, j = {1, 2}, i 6= j, and K1 + K2 ≤ r. If both suppliers have tested the components, Supplier i’s expected profit if releasing both components will be π ˜i (Ti , R1 , R2 ) =

∗ (T , T , R , R )−t−K . The expected profit in the case of θi θj (wi −ci )−(1−θi θj )(p+lsi +lb )−t = πm 1 2 1 2 i ∗ (T , T , nR , nR )−t−K . non-release of the component is π ˜i (Ti , nRi , Rj ) = −t−p0i = −t−Ki = πm 1 2 1 2 i

∗ (T , T , R , R )−t−K ≥ π ∗ (T , T , nR , nR )−t−K , Supplier i will release both components if πm 1 2 1 2 i 1 2 i m 1 2 ∗ (T , T , R , R ) ≥ π ∗ (T , T , nR , nR ). A similar logic holds if one or which is equivalent to πm 1 2 1 2 1 2 m 1 2

both suppliers have not tested the component. Hence, the release decision induced by the modified contract is equivalent to the first-best decision.

21

8.2.2

Testing Decision

∗ (T , T ) = 1 − If the integrated firm chooses to test both components, its expected profit is πm 1 2
∗ F12 (θ ) Eθ1 θ2 ≥θ∗ {θ1 θ2 (r − c1 − c2 ) − (1 − θ1 θ2 )(ls1 + ls2 + lb )} − 2t. To make the testing decision, the

firm needs to compare this expected profit to the expected h  profits from  testing only one or none of θ∗ ∗ the components, which are respectively πm (Ti , nTj ) = 1 − Fi θ Eθi ≥ θ∗ {θi θj (r − c1 − c2 ) − j θj i+
+ ∗ (1 − θi θj )(ls1 + ls2 + lb )} − t and πm (nT1 , nT2 ) = θ1 θ2 (r − c1 − c2 ) − (1 − θ1 θ2 )(ls1 + ls2 + lb ) . Hence, the integrated firm will make the following testing decisions:

∗ (T , T ) > π ∗ (nT , T ), π ∗ (T , T ) > π ∗ (T , nT ), and π ∗ (T , T ) > π ∗ (nT , nT ); • (T1 , T2 ) if πm 1 2 1 2 2 1 2 m m 1 2 m 1 m 1 2 m ∗ (nT , T ) > π ∗ (T , T ), π ∗ (nT , T ) > π ∗ (T , nT ), and π ∗ (nT , T ) > π ∗ (nT , nT ); • (nT1 , T2 ) if πm 1 2 1 2 2 1 2 1 2 m 1 2 m m 1 m m ∗ (T , nT ) > π ∗ (T , T ), π ∗ (T , nT ) > π ∗ (nT , T ), and π ∗ (T , nT ) > π ∗ (nT , nT ); • (T1 , nT2 ) if πm 1 2 2 1 2 2 1 2 m 1 2 m 1 m m 1 m

and ∗ (nT , nT ) > π ∗ (nT , T ), π ∗ (nT , nT ) > π ∗ (T , nT ), and π ∗ (nT , nT ) > • (nT1 , nT2 ) if πm 1 2 2 1 2 1 2 1 2 m m 1 m m ∗ (T , T ). πm 1 2

Now consider Supplier i’s expected profit under the modified contract as defined above. If Supplier i chooses to test the component, and Supplier j also tests the component, the ex
pected profit is π ˜si (Ti , Tj ) = 1 − F12 (θ∗ ) Eθ1 θ2 ≥θ∗ {θ1 θ2 (wi − ci ) − (1 − θ1 θ2 )(pi + lsi )} − t =
∗ (T , T ) + t − K . Now 1 − F12 (θ∗ ) Eθ1 θ2 ≥θ∗ {θ1 θ2 (r − c1 − c2 ) − (1 − θ1 θ2 )(lb + ls1 + ls2 )} − t − Ki = πm i j i consider the case in which Supplier ih tests thecomponent while Supplier j does not. The expected   i+ θ∗ profit for Supplier i is π ˜si (Ti , nTj ) = 1 − Fi θ Eθi ≥ θ∗ {θi θj (wi −ci )−(1−θi θj )(lsi +lb )}−t = j

∗ (T , nT ) πm i j

θj

− Ki . If Supplier i does not test the component, butSupplier j does test the compo h θ∗ Eθj ≥ θ∗ {θj θi (wi − ci ) − (1 − nent, the expected profit for Supplier i is π ˜si (nTi , Tj ) = 1 − Fj θ i θi i+ ∗ θj θi )(lsi +lb )} = πm (nTi , Tj )+t−Ki . Finally, if neither supplier tests the component, the expected
+ ∗ (nT , nT ) − K . profit of Supplier i is π ˜si (nT1 , nT2 ) = θ1 θ2 (wi − ci ) − (1 − θ1 θ2 )(lsi + lb ) = πm 1 2 i Now we summarize the above analysis by constructing the payoff matrix for the game between

suppliers at the testing stage in Table 1. Table 1: Suppliers’ Payoff Matrix HH S H 2 S1 HH H

T1 nT1

T2

nT2

∗ (T , T ) + t − K , π ∗ (T , T ) + t − K πm 1 2 1 m 1 2 2

∗ (T , nT ) − K , π ∗ (T , nT ) + t − K πm 1 2 1 m 1 2 2

∗ (nT , T ) + t − K , π ∗ (nT , T ) − K πm 1 2 1 m 1 2 2

∗ (nT , nT ) − K , π ∗ (nT , nT ) − K πm 1 2 1 m 1 2 2

Next we define the conditions for each set of strategies as a Nash equilibrium: ∗ (T , T ) > π ∗ (nT , T ) and π ∗ (T , T ) > π ∗ (T , nT ); the equilibrium is unique • (T1 , T2 ) if πm 1 2 1 2 2 m m 1 2 m 1 ∗ (nT , T ) > π ∗ (nT , nT ) or π ∗ (T , nT ) > π ∗ (nT , nT ); if πm 1 2 1 2 2 1 2 m m 1 m

22

∗ (nT , T ) > π ∗ (nT , nT ) and π ∗ (nT , T ) > π ∗ (T , T ); the equilibrium is • (nT1 , T2 ) if πm 1 2 1 2 1 2 m m m 1 2 ∗ (nT , nT ) > π ∗ (T , nT ) or π ∗ (T , T ) > π ∗ (T , nT ); unique if πm 1 2 2 2 m 1 m 1 2 m 1 ∗ (T , nT ) > π ∗ (nT , nT ) and π ∗ (T , nT ) > π ∗ (T , T ); the equilibrium is • (T1 , nT2 ) if πm 1 2 1 2 2 m m 1 m 1 2 ∗ (nT , nT ) > π ∗ (nT , T ) or π ∗ (T , T ) > π ∗ (nT , T ); and unique if πm 1 2 1 2 1 2 m m 1 2 m ∗ (nT , nT ) > π ∗ (nT , T ) and π ∗ (nT , nT ) > π ∗ (T , nT ); the equilibrium • (nT1 , nT2 ) if πm 1 2 1 2 1 2 2 m m m 1 ∗ (nT , T ) > π ∗ (T , T ) or π ∗ (T , nT ) > π ∗ (T , T ). is unique if πm 1 2 2 m 1 2 m 1 m 1 2

After incorporating the equilibrium uniqueness conditions, the constraints derived for the firstbest scenario follow from the constraints for the Nash equilibria detailed above. Accordingly, the modified contract achieves the first-best outcome.

9

Conclusion

This paper models supplier incentives to perform component testing prior to its release to the buyer for mass production. We have shown that simple contracts such as a reward or residual claimant contracts cannot ensure an efficient outcome, leading to component under- or over-testing by the supplier, which means the supplier either does not perform the essential tests, or performs them even when the cost of testing is excessive. Furthermore, the supplier may choose not to release the component when it is optimal for the supply chain to have it released or, conversely, choose to release it when she should not. Our model leads to several noteworthy managerial insights. First, for both the reward and residual claimant contract, we have found that if the supplier’s reward is high enough, any further increase in the reward makes the supplier less likely to test the component. Starting from a certain value, any higher reward creates a strong incentive for the supplier to release the component untested, while the value of testing does not increase as quickly in the reward. This dynamic reduces the supplier’s testing cost threshold. Furthermore, we have precisely identified the reward value after which any further increase in reward is detrimental to component testing. Exceeding this upper limit on the reward value leads to indifference on the supplier’s part with regard to whether to release the untested component or quit the project (i.e., neither testing nor releasing the component). This value is uniquely defined for both contract types. We have also established there is an efficient contract that leads to the first-best outcome. However, it requires the buyer to impose a penalty on the supplier even if the latter prefers not to release the component after performing the tests. Essentially, the efficient contract takes the form of an extended residual claimant contract that implies the additional penalty. In our model, it is the free exit option that creates inefficiencies in the supply chain. Eliminating this option allows the buyer to derive such a mix of the reward and penalty that the contract leads to an efficient outcome.

23

We will next discuss the limitations of our model as well as directions for future research. The model is relatively general as we do not assume any particular probability distributions for the component quality or the testing cost. Our choice to preserve generality comes at the cost of potentially obtaining more insights under more restrictive assumptions. An exciting opportunity for further research would be to derive the optimal parameters of various second-best contracts under typical probability distributions and compare their performance. A possible extension to the current research could be exploring the possibility of the buyer sharing the supplier’s testing costs. This could provide valuable insights on the desired degree of financial involvement for buyers when involving suppliers in product development. Another future research dimension could be to study the option of the parties improving the component once the testing results prove unsatisfactory. This could change the dynamic of the collaboration, making the supplier less likely to quit it when facing low component quality or other adverse circumstances. Thinking back to the grocery shopping example which opens this paper, we can now see that the shopper may sometimes behave in ways that are not in the best interests of the household. If there is no penalty for the dinner failure, the shopper is likely to mechanically follow the shopping list without thinking. Nor is it a good idea to inflate the reward for the dinner’s success to motivate the shopper to walk the extra mile. This will make them do exactly the opposite. However, implementing a penalty is not a solution either. If there is a penalty, the shopper is likely to double-check everything more rigorously than needed, irritate you by calling several times to clarify the list, and then they may still decide to abandon the shopping altogether and come home with a pizza instead. The best way to maximize the benefit to the household is to ensure that bringing home a pizza is not an easy risk-free solution. Accordingly, there needs to be a penalty for choosing the pizza option that is comparable to the penalty for the failed dinner.

Appendix Proof of Proposition 1 (a) If the tests have been performed, the component will be released if π ∗ (T, R) ≥ π ∗ (T, nR). Isolating the inequality for θ, we obtain θ ≥

ls +lb r−c+ls +lb .

Denoting

ls +lb r−c+ls +lb

If θ < θ∗ , then π ∗ (T, R) < π ∗ (T, nR), and the component is not released.

as θ∗ , we obtain θ ≥ θ∗ .

(b) If the tests have not been performed, the component will be released if π ∗ (nT, R) ≥

π ∗ (nT, nR). Isolating the inequality for θ, we obtain θ ≥

ls +lb r−c+ls +lb ,

or, equivalently, θ ≥ θ∗ .

If θ < θ∗ , then π ∗ (nT, R) < π ∗ (nT, nR), and the component is not released.

Proof of Proposition 2 (a) If π ∗ (nT, R) ≥ 0, this means π ∗ (nT ) = π ∗ (nT, R) = θ(r − c) − (1 − θ)(ls + lb ). Therefore, π ∗ (T ) ≥ π ∗ (nT ), Equation (4), takes the following form:


1 − F (θ∗ ) Eθ≥θ∗ {θ(r − c) − (1 − θ)(ls + lb )} − t ≥ θ(r − c) − (1 − θ)(ls + lb ) 24

(14)

Following the definition of the conditional expected value, we can rewrite (14) as ∗

1 − F (θ )




R1

θ∗


Z 1
θ(r − c) − (1 − θ)(ls + lb ) f (θ) dθ −t ≥ θ(r − c) − (1 − θ)(ls + lb ) f (θ) dθ (15) ∗ 1 − F (θ ) 0


R θ∗ Isolating (15) for t, we obtain t ≤ 0 (1 − θ)(ls + lb ) − θ(r − c) f (θ) dθ, and we denote
R θ∗ (1 − θ)(ls + lb ) − θ(r − c) f (θ) dθ as ta . Recall that (1 − θ)(ls + lb ) − θ(r − c) = 0 if θ = θ∗ . As 0

the integral is taken from 0 to θ∗ , then ta > 0.

Note that if t > ta , this means π ∗ (T ) < π ∗ (nT ), and thus the component will not be tested.

(b) If π ∗ (nT, R) < 0, this means π ∗ (nT ) = π ∗ (nT, nR) = 0. Therefore, π ∗ (T ) ≥ π ∗ (nT ),

Equation (4), takes the following form:


1 − F (θ∗ ) Eθ≥θ∗ {θ(r − c) − (1 − θ)(ls + lb )} − t ≥ 0

(16)

Isolating the above for t and using the definition of the conditional expected value, we obtain

R1 R1 t ≤ θ∗ θ(r − c) − (1 − θ)(ls + lb ) f (θ) dθ, and we denote θ∗ θ(r − c) − (1 − θ)(ls + lb ) f (θ) dθ as

tb . Recall that θ(r − c) − (1 − θ)(ls + lb ) = 0 if θ = θ∗ . As the integral is taken from θ∗ to 1, then tb > 0.

Note that if t > tb , this means π ∗ (T ) < π ∗ (nT ), and thus the component will not be tested.

Proof of Proposition 3 (a) If the tests have been performed, the component will be released if π ˆ (T, R) ≥ π ˆ (T, nR). Isolating ls ls ˆ ˆ If θ < θ, ˆ then the inequality for θ, we obtain θ ≥ . Denoting as θ, we obtain θ ≥ θ. w−c+ls

w−c+ls

π ˆ (T, R) < π ˆ (T, nR), and the component is not released.

(b) If the tests have not been performed, the component will be released if π ˆ (nT, R) ≥ π ˆ (nT, nR). ls ˆ If θ < θ, ˆ then Isolating the inequality for θ, we obtain θ ≥ , or, equivalently, θ ≥ θ. w−c+ls

π ˆ (T, R) < π ˆ (T, nR), and the component is not released.

Proof of Corollary 1 Solving θ∗ = θˆ for w, we obtain w = w > (<)

ls r+lb c ls +lb .

ls r+lb c ls +lb .

Furthermore, isolating θ∗ > (<) θˆ for w, we obtain

Proof of Proposition 4 (a) If π ˆs (nT, R) ≥ 0, this means π ˆs (nT ) = π ˆs (nT, R) = θ(w − c) − (1 − θ)ls . Therefore, π ˆs (T ) ≥

π ˆs (nT ) takes the following form:


ˆ E ˆ{θ(w − c) − (1 − θ)ls } − t ≥ θ(w − c) − (1 − θ)ls 1 − F (θ) θ≥θ

25

(17)

Following the definition of the conditional expected value, we can rewrite (17) as
ˆ 1 − F (θ)

R1 θˆ


Z 1
θ(w − c) − (1 − θ)ls f (θ) dθ −t≥ θ(w − c) − (1 − θ)ls f (θ) dθ ˆ 1 − F (θ) 0

(18)


R θˆ R θˆ Isolating (18) for t, we obtain t ≤ 0 (1 − θ)ls − θ(w − c) f (θ) dθ, and we denote 0 (1 − θ)ls −
ˆ As the integral is taken from θ(w − c) f (θ) dθ as tˆa . Recall that (1 − θ)ls − θ(w − c) = 0 if θ = θ. ˆ then tˆa > 0. 0 to θ, Note that if t > tˆa , this means π ˆs (T ) < π ˆs (nT ), and thus the component will not be tested.

(b) If π ˆs (nT, R) < 0, this means π ˆs (nT ) = π ˆs (nT, nR) = 0. Therefore, π ˆs (T ) ≥ π ˆs (nT ) takes

the following form:


ˆ E ˆ{θ(w − c) − (1 − θ)ls } − t ≥ 0 1 − F (θ) θ≥θ

(19)

Isolating the above for t and using the definition of the conditional expected value, we obtain

R1 R1 t ≤ θˆ θ(w − c) − (1 − θ)ls f (θ) dθ, and we denote θˆ θ(w − c) − (1 − θ)ls f (θ) dθ as tˆb . Recall ˆ As the integral is taken from θˆ to 1, then tˆb > 0. that θ(w − c) − (1 − θ)ls = 0 if θ = θ. Note that if t > tˆb , this means π ˆs (T ) < π ˆs (nT ), and thus the component will not be tested.

Proof of Proposition 5 We need to compare four testing cost thresholds: ta , tb , tˆa , and tˆb . First, we will show that at ˆs (nT, R) are always either w = ls r+lb c (at this value of w, θ∗ = θˆ from Corollary 1), π ∗ (nT, R) and π ls +lb

jointly non-negative or jointly negative. This will allow us to limit our comparison to two cases: ta to tˆa and tb to tˆb . h i bc − c) − (1 − Consider π ∗ (nT, R)−ˆ πs (nT, R)|w= ls r+lb c = θ(r−c)−(1−θ)(lb +ls )− θ( lslsr+l θ)l = s +lb lb (θ(r−c)−(1−θ)(lb +ls )) lb +ls

ls +lb

=

lb ∗ lb +ls π (nT, R).

Therefore, π ˆs (nT, R)|w= ls r+lb c = ls +lb

ls ∗ lb +ls π (nT, R).

We can

see that the two profits are always of the same sign, which allows us to compare the thresholds separately for cases (a) and the thresholds separately for cases (b) in Propositions 2 and 4. Now we compare ta to tˆa at w = ls r+lb c . It is easy to verify that (1 − θ)(ls + lb ) − θ(r − c) = (1 − ls +lb

θ)ls −θ(w−c)|w= ls r+lb c ,θ=θ∗ . Furthermore, (1−θ)(ls +lb )−θ(r−c) > (1−θ)ls −θ(w−c)|w= ls r+lb c ,θ<θ∗ . Therefore, ta =

Z

0

ls +lb

ls +lb

θ∗

((1 − θ)(ls + lb ) − θ(r − c)) f (θ) dθ > Z θ∗ ((1 − θ)ls − θ(w − c)) f (θ) dθ|w= ls r+lb c = tˆa |w= ls r+lb c ls +lb

0

Finally, we compare tb to tˆb at w =

ls r+lb c ls +lb .

(20)

ls +lb

It is easy to verify that θ(r−c)−(1−θ)(ls +lb ) = θ(w−

c)−(1−θ)ls |w= ls r+lb c ,θ=θ∗ . Furthermore, θ(r−c)−(1−θ)(ls +lb ) > θ(w−c)−(1−θ)ls |w= ls r+lb c ,θ>θ∗ . ls +lb

ls +lb

26

Therefore, tb =

Z

1

θ∗

(θ(r − c) − (1 − θ)(ls + lb )) f (θ) dθ > Z 1 (θ(w − c) − (1 − θ)ls ) f (θ) dθ|w= ls r+lb c = tˆb |w= ls r+lb c θ∗

ls +lb

(21)

ls +lb

Proof of Proposition 6 (a) If the tests have been performed, the component will be released if π ˜ (T, R) ≥ π ˜ (T, nR). Isolating ls +lb +K ls +lb +K ˜ we obtain θ ≥ θ. ˜ If θ < θ, ˜ the inequality for θ, we obtain θ ≥ r−c+ls +lb . Denoting r−c+ls +lb as θ, then π ˜ (T, R) < π ˜ (T, nR), and the component is not released.

(b) If the tests have not been performed, the component will be released if π ˜ (nT, R) ≥ π ˜ (nT, nR). ls +lb +K ˜ If θ < θ, ˜ then Isolating the inequality for θ, we obtain θ ≥ , or, equivalently, θ ≥ θ. r−c+ls +lb

π ˜ (T, R) < π ˜ (T, nR), and the component is not released.

Proof of Proposition 7 (a) If π ˜s (nT, R) ≥ 0, this means π ˜s (nT ) = π ˜s (nT, R) = θ(r − c) − (1 − θ)(ls + lb ) − K. Therefore, π ˜s (T ) ≥ π ˜s (nT ) takes the following form:


˜ E ˜{θ(r − c) − (1 − θ)(ls + lb ) − K} − t ≥ θ(r − c) − (1 − θ)(ls + lb ) − K 1 − F (θ) θ≥θ

(22)

Following the definition of the conditional expected value, we can rewrite (22) as


Z 1
θ(r − c) − (1 − θ)(ls + lb ) − K f (θ) dθ −t ≥ θ(r−c)−(1−θ)(ls +lb )−K f (θ) dθ ˜ 1 − F (θ) 0 (23)
R θ˜ Isolating (23) for t, we obtain t ≤ 0 (1 − θ)(ls + lb ) − θ(r − c) + K f (θ) dθ. Using the fact that
R θ∗ R θ˜ θ˜ ≥ θ∗ , we can rewrite it as t ≤ 0 (1−θ)(ls +lb )−θ(r −c)+K f (θ) dθ + θ∗ (1−θ)(ls +lb )−θ(r −

R θ˜ c) + K f (θ) dθ. It is equivalent to t ≤ ta + F (θ∗ )K + θ∗ K + (1 − θ)(ls + lb ) − θ(r − c) f (θ) dθ =
R ˜ + θ˜∗ (1−θ)(ls +lb )−θ(r −c) f (θ) dθ ≡ t˜a . Recall that (1−θ)(ls +lb )−θ(r −c)+K = 0 ta +F (θ)K θ ˜ As the integral is taken from 0 to θ, ˜ then t˜a > 0. if θ = θ.
˜ 1−F (θ)

R1 θ˜

Note that if t > t˜a , this means π ˜s (T ) < π ˜s (nT ), and thus the component will not be tested.

(b) If π ˜s (nT, R) < 0, this means π ˜s (nT ) = π ˜s (nT, nR) = 0. Therefore, π ˜s (T ) ≥ π ˜s (nT ) takes

the following form:


˜ E ˜{θ(r − c) − (1 − θ)(ls + lb ) − K} − t ≥ 0 1 − F (θ) θ≥θ

(24)

Isolating the above for t and using the definition of the conditional expected value, we obtain
R1 t ≤ θ˜ θ(r − c) − (1 − θ)(ls + lb ) − K f (θ) dθ. Using the fact that θ˜ ≥ θ∗ , we can rewrite it as

R1 R θ˜ t ≤ θ∗ θ(r − c) − (1 − θ)(ls + lb ) − K f (θ) dθ − θ∗ θ(r − c) − (1 − θ)(ls + lb ) − K f (θ) dθ. It 27



R θ˜ is equivalent to t ≤ tb − 1 − F (θ∗ ) K + θ∗ K + (1 − θ)(ls + lb ) − θ(r − c) f (θ) dθ = tb − 1 −

R ˜ K + θ˜∗ (1 − θ)(ls + lb ) − θ(r − c) f (θ) dθ ≡ t˜b . Recall that θ(r − c) − (1 − θ)(ls + lb ) − K = 0 F (θ) θ ˜ As the integral is taken from θ˜ to 1, then t˜b > 0. if θ = θ. Note that if t > t˜b , this means π ˜s (T ) < π ˜s (nT ), and thus the component will not be tested.

Proof of Proposition 8 If the component has been tested, the expected profits for the case of release and non-release are πs (T, R) = θ(r − c) − (1 − θ)(ls + lb ) − t − K and πs (T, nR) = −t − p0 = −t − K, respectively. If

the component has not been tested, πs (nT, R) = θ(r − c) − (1 − θ)(ls + lb ) − K and πs (nT, nR) =

−p0 = −K.

Next, we find the release threshold by solving (a) πs (T, R) ≥ πs (T, nR) and (b) πs (nT, R) ≥

πs (nT, nR). For both cases we find that θ (or θ) must be less than or equal to z, where z =

ls +lb r−c+ls +ls ,

which is identical to the condition from Proposition 1, i.e., z = θ∗ . Now we turn to the testing decision. The supplier chooses to test the component if πs (T ) ≥

πs (nT ), or equivalently,


1 − F (z) Eθ≥z {θ(r − c) − (1 − θ)(ls + lb ) − K} − t − F (z)K

≥ θ(r − c) − (1 − θ)(ls + lb ) − K

where z = θ∗ =

ls +lb r−c+ls +lb .

Simplifying (25), we obtain


+

, (25)



+ 1 − F (z) Eθ≥z {θ(r − c) − (1 − θ)(ls + lb )} − t − K ≥ θ(r − c) − (1 − θ)(ls + lb ) − K . (26)

Note that for K = 0, (26) is equivalent to (4). We can see that if (26) holds at K = 0, i.e., (4)
holds meaning π ∗ (T ) ≥ π ∗ (nT ), it will hold for all K ≤ 1−F (z) Eθ≥z {θ(r −c)−(1−θ)(ls +lb )}−t, or equivalently K ≤ π ∗ (T ).

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