Quality investment,inspection policy,and pricing decisions in a decentralized supply chain

Abstract

This paper studies the interaction between two key quality management decisions—input conformance quality and inspection policy—and related wholesale and retail prices in a two echelon supply chain. Market demand depends on the retail price as well as the end‐product conformance quality, which itself depends on the input quality and the inspection scheme. Consistent with previous empirical findings in the literature, we show that an increase in quality does not always result in higher prices for consumers due to the cost‐lowering effect of better quality. We also show that a lower input quality may still result in higher end‐product quality because of how it might incentivize more and/or better inspection. Any interaction between input quality and inspection policy becomes more pronounced in the decentralized system due to incentive asymmetry between the channel partners. This makes the adoption of a full‐inspection policy more likely there compared to an integrated system. Indeed, while vertical competition due to decentralization results in higher prices for customers, it can also result in better quality of end products. Another interesting finding in the decentralized setting is that, somewhat counterintuitively, a player may indeed opt to bear a higher share of the penalty for defective products sold to consumers resulting in higher profits for the player.

INTRODUCTION

In today's competitive market environment, product quality and price are two critical factors that shape the market demand for a product. In a survey on the automobile industry, consumers rated price and product quality as the two most important factors influencing over 60% of consumers' purchase decision (Deloitte, 2014). Quality's impact becomes even more consequential when we take into account that it also affects prices. For example, higher levels of quality, in addition to enhancing demand and potentially lowering costs, may also allow firms to charge higher prices (Dawar & Sarvary, 1997). In this paper, we focus on conformance quality, which refers to a product's ability to deliver on its stated specifications.

The conformance quality of the end product is affected by a number of factors. We focus on two of them—input conformance quality of suppliers and inspection technology and policy of buyers. A higher level of each of these two factors, independently, results in higher levels of end‐product quality. Typically, costs associated with input conformance quality and inspection are reflected in pricing by suppliers and/or buyers. More importantly, in a market where demand is driven by both price and quality, the interplay between these three factors is complex and unpredictable. The relation is further compounded in a decentralized supply chain due to the lack of alignment between decision makers. For example, the supplier's investment in improving the conformance quality may result in better input quality, but higher wholesale prices, for the buyer. As for the buyer's investment decision in inspection technology, while better inspection has its advantages (e.g., lower consumer return costs), it also increases the buyer's costs. It might also reduce the incentive for the supplier to invest in quality. The interaction among supplier's investment in input quality, wholesale pricing and the buyer's inspection policy, and how they impact the end‐item quality and price in the context of a decentralized supply chain has not been explored before in the literature. In this paper, we address this gap. Indeed, the interaction is of interest even in an integrated system; but, as there is a single decision maker making both quality management decisions and no wholesale prices are involved, the precise nature of the interaction might be different in this context. So, we also analyze the centralized system and compare how the interaction is affected by the structure of the supply chain—decentralized or integrated.

To do so, we consider a framework that is an abstraction of reality faced by companies in sectors such as aerospace, automobile, and electronics. Specifically, our bilateral monopoly model consists of a single buyer (e.g., an aerospace engine manufacturer or an electronics retailer) buying from a single supplier and selling to end consumers. Given the nature of the industry, consumer demand depends on the quality as well as the price. Information about end products not conforming to specifications (e.g., overheating for engines or electronics) is publicly available to the end customers (e.g., from National Transportation Safety Board for engines and www.consumerreports.org for electronics). In the paper, we focus on products for which the end‐product's conformance quality depends critically on the supplier's input quality investment in a key component, for example, the material used, the tests done, the processes used, etc. Examples of such sub‐assemblies for an aircraft engine would be fuel control and bearing assembly, while for electronics can be battery packs. Along with quality investment decision, the supplier also sets the wholesale price for the product.

The main role that the buyer plays in our setting is inspecting the product and deciding on the retail price. The overall end‐product conformance quality is clearly dependent on the supplier's input quality in the key component and the buyer's inspection policy. In the context of inspection, the buyer needs to make two decisions—the choice of the inspection accuracy technology (along with its associated cost), and what fraction of the supply to inspect— given the input quality of the supplier. In line with reality, we assume that the supplier has to replace any item found defective during the buyer's inspection process, and also bears the associated cost. Subsequently, the buyer decides on the resale price to charge to the customers. As the inspection process may not be fully accurate and/or the buyer does not necessarily perform full inspection, defective items, that is, items not conforming to specifications, can still end up in the market. We assume that all such defective items will be returned by customers. Defective items on the field may result in significant costs due to service/replacement (aerospace, automobile), and loss of customer goodwill (electronics). These external failure costs are typically shared by both channel partners (see Reyniers & Tapiero, 1995b; Wan & Xu, 2008). One example of external failure cost sharing in the electronics industry is when Sanyo (supplier) and Lenovo (buyer) shared the $17 million cost of recalling 205,000 Sanyo made laptop battery packs that could overheat due to a flaw in product design in 2007 (Nystedt, 2007). Similarly, Ford and Firestone shared the cost associated with recall of tires made by Firestone and installed in Ford vehicles (Chao et al., 2009). Most recently, LG agree to reimburse GM $1.9 billion for costs from battery‐related fires in Chevy Bolt EVs (Yang, 2021). The basic framework remains the same in an integrated system except that there are no wholesale prices or cost sharing involved and all quality investment decisions are made by tone entity.

We build on the above framework to study the joint effect of quality, inspection, and pricing decisions in both decentralized and integrated systems. As end customers are quality and price sensitive, the relative strength of these factors determines the outcomes in both systems. Consequently, we note the possibility of two interesting outcomes. First, higher quality for end consumers does not necessarily result in higher prices for them providing analytical support to the empirical finding that quality–price relationship is not necessarily positive (Dawar & Sarvary, 1997). We provide explanation for this by examining the opposing cost‐reducing and revenue‐enhancing effects of higher quality. Second, higher quality cost does not necessarily result in low‐quality product for end consumers due to the moderating effect of better and/or more inspection. We are able to demonstrate under what conditions the above phenomenon is likely to happen so that the supply chain offers a better quality end product even when input quality is more expensive, and vice versa. In the comparison between the integrated and decentralized systems, we show that inspection is more likely in the decentralized system, as input quality is lower due to the double marginalization effect. This leads to the buyer being more likely to use inspection. As such, vertical competition in a decentralized system can indeed lead to higher quality end products, even though consumers might have to pay higher prices for them. Finally, we study the effect of sharing of external failure costs between decentralized supply chain partners, and show that the supplier (respectively, buyer) may be willing to share a higher fraction, that is, absorb higher risk, due to the resulting impact on the buyer's (respectively, supplier's) inspection (respectively, input quality) policy.

LITERATURE REVIEW

In one of the earlier papers, Mussa and Rosen (1978) studied the joint impact of quality and price in a monopolistic setting. Later, their model was extended to settings with horizontal and vertical competitions by Moorthy (1988). The extant literature has analyzed the impact of price and nonprice factors such as quality on demand (Bernstein & Federgruen, 2004; Desai, 1997; Gurnani et al., 2007; Iyer, 1998). In recent years, contracting mechanisms such as revenue‐sharing contracts (Ouardighi & Kim, 2010; Liu et al., 2016), recall cost‐sharing contracts (Chao et al., 2009), and return strategies (Li et al., 2019) have been introduced to coordinate supply chains in the context of quality decisions. However, the above papers do not include the buyer's inspection decision that critically affects quality and pricing decisions, as is evident from our paper. The quality–price relationship has also been studied empirically in the marketing literature with results suggesting it to be product‐specific (Morris & Bronson, 1969), weak (Gardner, 1971) and can be positive or negative depending on the circumstances (Dawar & Sarvary, 1997).

The interaction of inspection policy with input quality investment has attracted the attention of quite a few researchers. In this context, inspection policy typically involves deciding on the fraction of inbound products to be inspected and in few cases, on inspection accuracy decision. The related work vary in terms of their assumptions about the order in which inspection policy and input quality are committed. These include cases where they are decided simultaneously (Reyniers & Tapiero, 1995b, 1995a), or the inspection policy is determined before the input quality (Babich & Tang, 2015; Balachandran & Radhakrishnan, 2005; Chen et al., 2022; Hwang et al., 2006; Lim, 2001; Starbird, 2001; Wan & Xu, 2008), or the input quality is committed before the inspection policy is set (Baiman et al., 2000; Hsieh & Liu, 2010; Lee & Rosenblatt, 1985). Although some papers assume exogenous inspection accuracy that may be perfect (Babich & Tang, 2015; Lee & Rosenblatt, 1985; Lim, 2001; Reyniers & Tapiero, 1995a, 1995b; Starbird, 2001) or imperfect (Babich & Tang, 2015; Balachandran & Radhakrishnan, 2005; Hsieh & Liu, 2010; Hwang et al., 2006; Wan & Xu, 2008), some others, like us, incorporate inspection accuracy as a decision variable (Baiman et al., 2000; Lee & Li, 2018).

Assuming all‐or‐nothing inspection options, Reyniers and Tapiero (1995b, 1995a) identify pure and mixed strategy equilibria based on the internal failure penalties and how channel partners share the external failure costs, and propose channel‐coordinating contracts. In a buyer‐leading setting, Lim (2001) considers a contract that is offered by the buyer in presence of information asymmetry regarding supplier's product quality, which is exogenous. The contract specifies the probability that the buyer will inspect the product, rebate amount for defective items, and penalties for returns from the end customer. The supplier responds by simply accepting or rejecting the contract. Starbird (2001) investigates the impact of a buyer's penalty and reward terms on the supplier's input quality under three inspection regimes; no inspection, full inspection and sampling. Wan and Xu (2008) consider a Stackelberg setting where the buyer moves first and commits to the number of items to inspect and the supplier responds with his input quality decision. Any defective item identified by the inspection process is reworked by the supplier and the external failure cost is shared by both parties. The authors show that a fractional inspection policy or an all‐or‐nothing inspection policy can be the equilibrium outcome, depending on the supplier's share of the failure cost. In our work, in contrast to the above papers, the buyer's decision about the proportion of items to inspect follows the supplier's input quality decision.

In an early paper, Lee and Rosenblatt (1985) studied the buyer's optimal inspection policy given the supplier's input quality. Their analysis shows that full inspection is optimal when the input quality is neither too high nor too low. Otherwise, inspection is not a preferable option for the buyer. More recently, Hsieh and Liu (2010) consider the case where both the supplier and the buyer invest in quality and inspection. They study the interaction of the supplier and the buyer under this setting for given market prices, penalties, and post‐sale costs. In these papers, the inspection accuracy is assumed to be exogenous, whereas we include accuracy as an endogenous variable. Baiman et al. (2000) consider endogenous inspection accuracy in their work where they analyze the relation between product quality, the cost of quality, and the information that can be contracted upon. In their setting, the supplier moves first and chooses a quality level for the product that is equivalent to the probability that the product is good. In response, the buyer invests in inspection accuracy to appraise the quality of the product. Lee and Li (2018) also consider inspection accuracy as an endogenous decision made by the buyer. In contrast to Baiman et al. (2000), they study a setting where the buyer moves first and offers a contract to the supplier that stipulates a wholesale price and an internal failure penalty for the supplier. If the supplier accepts the contract, both parties simultaneously decide on their efforts on quality and in addition, buyer sets her inspection accuracy level. In their model, the resale price is assumed to be exogenous. The authors observe that with efforts exerted by both parties, when the external failure can be contracted by the buyer, the equilibrium results in first‐best outcome.

Although the above‐mentioned papers study the relation between input quality and inspection, in contrast to us, resale prices and demand are assumed exogenous. More importantly, inspection accuracy in the majority of the above literature is also exogenous. Balachandran and Radhakrishnan (2005) consider inspection accuracy as an investment decision in their model. However, they consider a setting where the wholesale price is determined by the buyer subject to the supplier's participation constraint. In a following study, Hwang et al. (2006) consider a problem where the supplier's quality is unobservable and both parties can perform inspection. They analyze two strategies in the paper—the appraisal regime with traditional inspection option, and the certification regime to induce the supplier's high quality. But, their focus is different as demand is exogenous. Babich and Tang (2015) study the role of using deferred payments and compare it to using inspection to manage product adulteration. They find four factors affecting the choice of the dominating strategy: inspection cost and accuracy, buyer's liability, difference in financing rates for the two parties, and the difference in production costs for adulterated and unadulterated product. Finally, Chen et al. (2022) compare two strategies to manage product quality: inspection‐based prevention and insurance‐based mitigation. They conclude that when the buyer's working capital is low, the benefit of inspection outstrips the benefit of insurance. The reverse is true beyond a threshold.

In summary, the extant literature either focuses on the interaction between the buyer's inspection policy and the supplier's input conformance quality decision, or on the impact of the quality investment decisions on the pricing policies. Therefore, studying the joint interaction across the three types of decisions (quality investment, inspection policy, and pricing) as well as understanding how this interaction is affected by the supply chain structure (integrated vs. decentralized) distinguishes our work and makes a novel contribution to the literature.

MODEL SETTING

Our basic model setting is a decentralized supply chain consisting of a supplier selling a product to a buyer who then sells it to customers. The end customer's demand function D is decreasing in the selling price, p, and increasing in the quality of the end product, θ. Specifically, θ represents the overall probability that a product received by the end consumer conforms to its stated specifications. As we assume an ergodic process, θ also represents the proportion of finished products that are nondefective. Therefore,

$0<\theta \le 1$

, and the value of θ is assumed to be public information. As indicated in Section 1, for products like airplanes, electronics retailers and automobiles end customers are indeed aware of such (non)conformance information and quality acceptance ratios tend to be quite high. For example, Astronics Armstrong Aerospace has a 92% quality acceptance ratio (Astronics, 2020), UTC Aerospace Systems has 95% (UTC, 2018), and Esterline Advanced Sensors requires 99.9% (Esterline, 2009) quality acceptance ratios from its suppliers. We model the demand expression as a function of the price and end‐product quality as follows:

$$D=\alpha -\beta p+\lambda \theta .$$

1

In the above demand function,

$(\alpha +\lambda )$

is the potential demand from customers who will buy the end product if it is known to be defect‐free (

$\theta =1$

), that is, always conform to the stated specifications. Moreover, λ represents the portion of the demand that is sensitive to conformance quality. Some of these customers will make their purchase decisions based on the defect rate of the product which is public information. As such, lower θ (more nonconforming end products) results in higher proportion of the λ customers not buying the product. Note that even though nonconforming, defective items sold to end customers are replaced, as the replacement process would lead to delayed use of the product by the customer and also result in inconvenience, we have modeled this disutility by assuming the above property of the demand function. On the other hand, β (>0) is the price sensitivity of demand. Similar demand models have been used by Desai (1997), Bhardwaj (2001), Bernstein and Federgruen (2004), and Gurnani et al. (2007). A defining characteristic of our paper is that the end‐product conformance probability θ depends on the strategic decision by the supplier about investment in input conformance quality and the buyer about her inspection technology/policy. We discuss them in detail below. Throughout the rest of the paper, any reference to quality will represent conformance quality, unless otherwise stated.

We represent the supplier's input conformance quality by q denoting the fraction of the conforming products. This is in line with related operations management literature (e.g., Anupindi & Akella, 1993; Tomlin, 2006). The buyer can observe q through samples provided in advance of the actual transaction. We reiterate that, in our context, the supplier's component plays a crucial role in the performance of the end product with the buyer's role primarily being inspection. Moreover, q also depends on supplier's quality related investments (e.g., in material, manpower, process, equipment) where the investment cost is given by

$S(q)$

, an increasing function of q. Given the conformance quality level, q, and the wholesale price, w, of the supplier's product, the buyer decides on the fraction of inbound products, U, to inspect (

$0\le U\le 1$

), the inspection accuracy option, ϕ, and the price of the end product, p. Clearly,

$U=0$

is the no‐inspection policy,

$0<U<1$

is the partial inspection policy, and

$U=1$

is the full inspection policy. To facilitate exposition and provide cleaner insights, without loss of generality, we assume that the buyer has two discrete inspection technology options: high accuracy (H) and low accuracy (L). Here, inspection accuracy measures the probability that a defective item can be correctly detected. High (respectively, low) accuracy option offers an inspection accuracy level of

${\varphi }_H$

(respectively,

${\varphi }_L$

) with a unit inspection cost of

${\psi }_H$

(respectively,

${\psi }_L$

). To avoid trivial cases, we assume that

${\varphi }_H>{\varphi }_L$

and

${\psi }_H>{\psi }_L$

. Moreover, inspection never identifies a good item as defective. As there is cost associated with inspecting each unit, the buyer may choose to inspect a fraction of the lot. However, we will show that at equilibrium, the buyer's best strategy is always either no inspection or full inspection. As such, U can be restricted to be either 0 or 1. Consequently, the buyer eventually faces three distinct inspection strategies: no inspection, full inspection with low accuracy, or full inspection with high accuracy.

As pointed out earlier, the end‐item conformance probability is determined by the supplier's input quality and the buyer's inspection strategy. Without inspection, the end‐item no‐defect probability will be directly equal to the supplier's input conformance quality (i.e.,

$\theta =q$

). On the other hand, with full inspection,

$\theta =q+(1-q)\varphi$

. Any item found to be defective during the buyer's inspection process is returned to the supplier for rework. It is assumed that all reworked items are nondefective. The rework process incurs an additional cost of v per unit for the supplier, which is referred to as the internal failure cost. This process includes all shipment, remanufacturing, and/or repair expenses. Hence, in general, the end‐product quality is

$\theta =q+(1-q)U\varphi$

. Clearly, with full inspection and perfect accuracy θ converges to 1. Similar representations of end‐item quality are employed in the operations management literature (e.g., Lim, 2001; Wan & Xu, 2008).

Any defective item sold in the market is returned by the end customers. They are then reworked and delivered back to the customer. We assume that every reworked item is nondefective. This assumption is consistent with the extant operations management literature. Examples include Lim (2001), Wan and Xu (2008), Lu et al. (2009), Chao et al. (2009), and Xiao et al. (2011). Any returned item prompts an external failure cost caused by customer's loss of goodwill. This cost, denoted by g, is additional to the internal failure cost incurred by the supplier, who is responsible to rework the product. As such, the cost of selling a defect item to the end consumer amounts to

$g+v$

. The external failure cost, g, is shared by both parties. Specifically, the supplier bears

$\gamma g$

while the buyer is charged

$(1-\gamma )g$

, where

$\gamma \ge 0$

and is referred to as the external failure cost share for the supplier. As discussed before, across many industries, including automobile, such cost‐sharing penalty contracts are common and is becoming more entrenched due to bad press associated with squabbles about recall cost sharing after high‐profile disasters (Purchasing, 1998). For

$\gamma >1$

, the supplier not only fully compensates the cost for returns but also pays a penalty to the buyer for each defective item sold to the end consumer. We focus on the interesting case where

$g>{\psi }_H$

, that is, the cost of external failures is higher than internal ones (as in Nagar & Rajan, 2001; Wan & Xu, 2008). The unit production cost is denoted by c, where we assume that

$v>c$

, that is, the marginal impact of q on unit production cost is less than the internal failure cost so that the supplier does not have any incentive to produce only defective items. Moreover, to ensure nonnegative prices and profits, we assume that

$\alpha \ge \beta (c+{\psi }_H+g+v)$

always holds.

We study the above general setting in two stages. In the first stage, the supplier sets her input conformance quality (q) and the wholesale price (w). In the second stage, the buyer responds by choosing her inspection accuracy level (ϕ), inspection policy (U), and resale price (p). In the basic setting, the external failure cost share (γ) is treated as an exogenous parameter. Later, in Subsection 5.3, we examine the impact of the γ as a decision variable on the strategies of the supply chain partners.

To understand how the investments in quality management decisions are affected by supply chain structures, we also study this problem under the integrated supply chain structure. In this case, there is a single decision maker for the chain and because there is no vertical competition, transfer payments (i.e., w) and the cost‐sharing parameter (i.e., γ) are not relevant. Moreover, the internal failure cost v is now incurred by the chain, rather than charged to the supplier. To establish the benchmark, we begin our analysis with the integrated system in the next section (Section 4). In Section 5, we analyze the decentralized setting and derive the equilibrium outcomes. Section 6 compares the results obtained for the integrated system and the decentralized supply chain, and discuss managerial insights obtained from this comparison. Finally, we conclude with Section 7.

ANALYSIS OF THE INTEGRATED SYSTEM

We first study the interaction between quality investment, inspection, and pricing strategy in the integrated system to establish a benchmark for our analysis of the decentralized supply chain in the subsequent sections. Recall that, in our setting, inspection policy and pricing decisions follow the input quality decision. The profit function for the integrated firm is given by:

$$\begin{array}{c}\hfill {\mathrm{\Pi }}_I=(p-c-U\psi -(1-q)v-(1-\theta )g)(\alpha -\beta p+\lambda \theta )-\frac{1}{2}s{q}^2.\end{array}$$

2

The profit for the integrated firm is calculated by subtracting the following cost components from the sales revenue: production cost, inspection cost, rework/repair cost, and the customer dissatisfaction cost. We note that the internal failure cost is independent of the inspection policy because all defective items are eventually reworked. As part of the overall optimal solution for the integrated system, we first make a couple of key observations. ¹ Lemma 1

For a given inspection technology, that is, (

$\varphi ,\psi$

) pair, there is a unique threshold quality investment level

$q_t^I=1-\frac{\beta \psi }{\varphi (\lambda +\beta g)}$

such that:

• (i)

  If

  $q<q_t^I$
  , then the integrated firm should: (a) inspect all items, that is,
  $U=1$
  , and (b) charge the following price to the end customers:
  $$p_{fI}^{*}=\frac{\alpha +\lambda (q+(1-q)\varphi )+\beta (c+\psi +(1-q)v+(1-\varphi )(1-q)g)}{2\beta }.$$

  3

• (ii)

  Otherwise, that is, if

  $q\ge q_t^I$
  , then it should: (a) perform no inspection, that is,
  $U=0$
  , and (b) the end customer price charged should be:
  $$p_{nI}^{*}=\frac{\alpha +\lambda q+\beta (c+(1-q)(v+g))}{2\beta }.$$

  4

The optimal profit for the integrated firm is deduced by substituting the optimal U and p pairs in (2).

The above result indicates that the best response to a given input quality is the all‐or‐nothing inspection policy. This is consistent with previous work that report similar conclusions for settings where inspection policy selection does not precede commitment to input quality (Deming, 2000; Lee & Rosenblatt, 1985). The threshold value given in the above proposition implies that the integrated firm is more likely to inspect when the demand is more sensitive to conformance quality (i.e., high λ), and less likely to inspect when the customers are highly price sensitive (high β). We observe that the optimal inspection policy choice is independent of the internal failure cost, v. This is expected because this cost is incurred regardless of whether inspection is employed or not. We note that the threshold quality level is between 0 and 1 only if

$\beta \psi <\varphi (\lambda +\beta g)$

implying that inspection is never an option if this inequality does not hold. Clearly, in that case inspection cost is too high to economically justify the return on improvement in quality output. To focus on only the interesting cases, we assume that the above inequality holds in the rest of the paper.

Substituting optimal inspection policy and price, the profit function for the integrated system becomes:

5

where

$$K(q)=\alpha +\lambda q-\beta \left(c+(1-q)(v+g)\right)$$

6

$$\text{and}\mathrm{\Delta }(q,\varphi ,\psi )=\varphi (1-q)(\lambda +\beta g)-\beta \psi .$$

7

It is easy to observe that no‐inspection profit function is strictly concave in q if and only if

$s>{(\lambda +\beta (v+g))}^2/(2\beta )$

. In the opposite case, the profit function is convex increasing at

$q=0$

implying that the optimal input quality must always be perfect (i.e.,

$q=1$

). In general, it is straightforward to deduce from the first‐order optimality condition for (5) that the optimal input quality is perfect if and only if

$s\le \widehat{s}$

, where

$$\widehat{s}=\frac{(\alpha +\lambda -\beta c)(\lambda +\beta (v+g))}{2\beta }.$$

8

Under this condition, as the input quality is perfect, there will be no need for inspection and the integrated firm does not incur any internal or external failure costs. We note that the bang–bang solution due to Lemma 1 pertaining to the inspection policy depends on whether the input quality of the product is above or below a threshold, which is a function of the inspection accuracy and the unit inspection cost. As such, each inspection technology option may result in a unique and different quality threshold. Let

$q_t^I(i)$

denote the quality threshold generated by the inspection option i (

$i=H,L$

). Our next result compares the quality thresholds for the two inspection technology options. Lemma 2

$q_t^I(H)\ge q_t^I(L)$

if and only if

${\psi }_H/{\varphi }_H\le {\psi }_L/{\varphi }_L$

.

The proof is straightforward from the comparison of

$q_t^I$

evaluated at

$\psi ={\psi }_i$

and

$\varphi ={\varphi }_i$

for

$i=H,L$

. The above result indicates that as input quality decreases, the optimal inspection policy switches from no inspection to full inspection at a higher accuracy level if the unit inspection cost at that level is sufficiently low, that is, the ratio of unit inspection cost to the accuracy level is lower than the other option. In general, the lowest cost to accuracy ratio option produces the highest quality threshold. Lemmas 1 and 2 indicate that if the input quality is above a certain threshold, no inspection is the optimal policy. For unit quality below the threshold, full inspection becomes optimal, but the choice of high or low inspection accuracy depends on the conditions outlined in Lemmas 1 and 2, and Equations (6) and (7). Moreover, we also know from concavity of the objective function that the optimal quality choice decreases with the quality cost coefficient s. Consequently, we can conclude that there is a unique mapping between the quality thresholds and the quality cost coefficient under optimality conditions. These observations lead us to the following fundamental result regarding the integrated firm's overall optimal strategy. Proposition 1

For given inspection technology choices (

${\varphi }_H,{\psi }_H$

) and (

${\varphi }_L,{\psi }_L$

),

• (i)

  if

  ${\psi }_H/{\varphi }_H\le {\psi }_L/{\varphi }_L$
  , there exists a unique quality cost threshold,
  $s_t^I(H)$
  , such that if
  $s\le s_t^I(H)$
  , then the integrated firm's optimal decision is not to perform any inspection. On the other hand, if
  $s>s_t^I(H)$
  , then the integrated firm's optimal decision is to perform full inspection with inspection option (
  ${\varphi }_H,{\psi }_H$
  ). In this case, inspection option (
  ${\varphi }_L,{\psi }_L$
  ) is never employed;

• (ii)

  if

  ${\psi }_H/{\varphi }_H>{\psi }_L/{\varphi }_L$
  and
  $\beta ({\psi }_H-{\psi }_L)\ge (\lambda +\beta r)({\varphi }_H-{\varphi }_L)$
  , there exists a unique quality cost threshold,
  $s_t^I(L)$
  , such that if
  $s\le s_t^I(L)$
  , then the integrated firm's optimal decision is not to perform any inspection. On the other hand, if
  $s>s_t^I(L)$
  , then the integrated firm's optimal decision is to perform full inspection with inspection option (
  ${\varphi }_L,{\psi }_L$
  ). In this case, inspection option (
  ${\varphi }_H,{\psi }_H$
  ) is never employed;

• (iii)

  if

  ${\psi }_H/{\varphi }_H>{\psi }_L/{\varphi }_L$
  and
  $\beta ({\psi }_H-{\psi }_L)<(\lambda +\beta r)({\varphi }_H-{\varphi }_L)$
  , there are two quality cost thresholds, namely,
  $s{s}_t^I(L)$
  and
  $s_t^I(HL)$
  such that if
  $s\le s_t^I(L)$
  , then the integrated firm's optimal decision is not to perform any inspection. If
  $s_t^I(HL)\ge s>s_t^I(L)$
  , then the integrated firm's optimal decision is to perform full inspection with inspection option (
  ${\varphi }_L,{\psi }_L$
  ). Finally, if
  $s>s_t^I(HL)$
  , where
  $s_t^I(HL)\ge s_t^I(H)$
  , the integrated firm's optimal decision is to perform full inspection with inspection option (
  ${\varphi }_H,{\psi }_H$
  ).

Based on the optimal inspection strategy, the optimal quality investment and price strategy can be deduced from Lemma 1 and shown in Table 1.

TABLE 1

Optimal input quality, price, and profit for the integrated system

No inspection	Full inspection (with ( ${\varphi }_i,{\psi }_i$ ))
$q_n^I={\displaystyle \frac{(\alpha -\beta (c+v+g))(\lambda +\beta (v+g))}{2\beta s-{(\lambda +\beta (v+g))}^2}}$	$q_f^I={\displaystyle \frac{(\alpha +{\varphi }_i\lambda -\beta (c+{\psi }_i+v+(1-\varphi )g))((1-{\varphi }_i)\lambda +\beta (v+(1-{\varphi }_i)g))}{2\beta s-{((1-{\varphi }_i)\lambda +\beta (v+(1-{\varphi }_i)g))}^2}}$
$p_n^I={\displaystyle \frac{\alpha +\beta (c+v+g)}{2\beta s-{(\lambda +\beta (v+g))}^2}}s$	$p_f^I={\displaystyle \frac{\alpha +{\varphi }_i\lambda +\beta (c+{\psi }_i+v+(1-\varphi )g)}{2\beta s-{((1-{\varphi }_i)\lambda +\beta (v+(1-{\varphi }_i)g))}^2}}s$
$D_n^I={\displaystyle \frac{\alpha -\beta (c+v+g)}{2\beta s-{(\lambda +\beta (v+g))}^2}}\beta s$	$D_f^I={\displaystyle \frac{\alpha +{\varphi }_i\lambda -\beta (c+{\psi }_i+v+(1-{\varphi }_i)g)}{2\beta s-{((1-{\varphi }_i)\lambda +\beta (v+(1-{\varphi }_i)g))}^2}}\beta s$
${\mathrm{\Pi }}_n^I={\displaystyle \frac{{(\alpha -\beta (c+v+g))}^{2}s}{2(2\beta s-{(\lambda +\beta (v+g))}^2)}}$	${\mathrm{\Pi }}_f^I={\displaystyle \frac{{(\alpha +{\varphi }_i\lambda -\beta (c+{\psi }_i+v+(1-{\varphi }_i)g))}^{2}s}{2(2\beta s-{((1-{\varphi }_i)\lambda +\beta (v+(1-{\varphi }_i)g))}^2)}}$
${\theta }_n^I=q_n^I$	${\theta }_f^I=q_f^I+{\varphi }_i(1-q_f^I)$

The above result suggests that when the unit inspection cost for the high‐accuracy mode is relatively cheap, it is the only preferable option when inspection is needed. At the other extreme, if the unit inspection cost for the high‐accuracy mode is quite expensive, it is dominated by the low‐accuracy option. When the cost is neither low nor high, the optimal strategy switches from low‐accuracy inspection to high‐accuracy inspection as the cost of quality increases (Figure 1). We note that beyond the cost thresholds end‐item quality depends on the inspection accuracy. With sufficiently high accuracy, end‐item quality may increase when the inspection regime switches from one inspection regime to the other as exemplified in Figure 1b. Because of this, indeed it is possible that end‐item quality is not strictly decreasing in the cost of input quality and might be nonmonotone. In general, threshold cost decreases as the inspection accuracy increases.

FIGURE 1

Input quality cost and end‐item quality

The optimal outcomes for the integrated system are summarized in Table 1. Although the mathematical expressions are cumbersome, closed‐form equations pertaining to threshold values for input quality cost can be derived from the optimal profit functions given in the table.

ANALYSIS OF THE DECENTRALIZED SUPPLY CHAIN

Clearly, the equilibrium outcome in the competitive decentralized supply chain will not be aligned with the optimal outcome obtained under the integrated system. In addition to double marginalization, the players' best strategy choices will be influenced by the internal failure cost (v), wholesale price (w), and how the external failure cost, γ, is shared between the two players, where γ represents the supplier's share of the external failure cost. Employing backward induction, we begin our analysis with the decision problem of the buyer, who responds to the supplier's input quality and wholesale price decisions. Later, we solve the supplier's decision problem based on the best response mapping of the buyer.

The Buyer's inspection policy and pricing

The buyer's decisions include her inspection accuracy level (ϕ), inspection policy (U), and resale price (p). Her profit function is given by the following equation:

$${\mathrm{\Pi }}_B^{}=\left(p-w-U\psi -(1-\theta )(1-\gamma )g\right)(\alpha -\beta p+\lambda \theta ).$$

9

The profit for the buyer is calculated by subtracting the following cost components from the sales revenue: procurement cost, inspection cost, and the buyer's portion of the external failure cost. We note that rework/repair cost is incurred by the supplier and as such not included in the buyer's profit function. To facilitate the analysis and presentation, we first derive the buyer's inspection policy and pricing decision for a given inspection accuracy level. The buyer's optimal strategy can be summarized as follows. Proposition 2

There is a unique threshold quality level

$q_t^B=1-\frac{\beta \psi }{\varphi (\lambda +\beta (1-\gamma )g)}$

such that:

• (i)

  if

  $q\le q_t^B$
  , the buyer always prefers full inspection (
  $U=1$
  ) and her optimal resale price is:
  $$p_{fB}^{*}=\frac{\alpha +\lambda (q+(1-q)\varphi )+\beta (w+\psi +(1-\varphi )(1-\gamma )(1-q)g)}{2\beta };$$

  10

• (ii)

  otherwise, that is, if

  $q>q_t^B$
  , the buyer prefers no inspection (
  $U=0$
  ) and the resale price with this policy is:
  $$p_{nB}^{*}=\frac{\alpha +\lambda q+\beta (w+(1-\gamma )(1-q)g)}{2\beta }.$$

  11

Similar to the integrated system, inspection policy selection is a bang–bang solution that is based on the input quality level. The intuition behind the all‐or‐nothing outcome is explained by the fact that the input defect rate is known, and as such, there is no need for learning or updating the conditional defect rates through sampling (Yao & Zheng, 2002). Different from the integrated system, the buyer's input quality threshold depends on her share of the external cost. Under the special case, where

$\gamma =0$

, that is, the buyer is fully responsible for the external failure cost, the quality threshold is identical to that of the integrated channel. In that case, as the quality thresholds are independent of the wholesale price and the internal failure cost, they are aligned across both channel structures. Consequently, we can observe that the quality threshold is always higher in the integrated system compared to the competitive setting. This is intuitive in that the higher margins enjoyed by the integrated firm allows for more room for the inspection option with a given quality level.

It is straightforward to see that

$q_t^B$

decreases in cost‐sharing parameter γ and the variable cost of inspection, ψ. As the supplier's external failure cost share and/or the cost of inspection increases, the buyer will have less incentive to inspect and as such will opt for inspection at relatively lower quality levels. It is also intuitive to observe that the threshold increases in demand enhancing impact of quality, λ, and the cost of external failure, g. As the inspection accuracy, ϕ, increases, inspection becomes more effective and cost‐efficient. As such, the threshold increases with ϕ. We also note that the quality threshold decreases in price sensitivity, β. Clearly, as the demand for the item becomes more sensitive to price, the quality will have relatively less weight on final sales disincentivizing inspection for the buyer. The buyer will never employ inspection with a (ψ, ϕ) pair when the supplier's external cost share is above a threshold indicated by the following inequality, as the quality threshold is below zero in this case:

$$\gamma \ge 1-\frac{\beta \psi -\varphi \lambda }{\varphi \beta g}.$$

12

In this case, the cost of inspection outstrips its benefits on gains in external failure costs and enhancement in demand volume. To avoid such cases, we assume that the supplier's cost share is below the threshold given above for both inspection accuracy options. Finally, it is interesting to observe that the buyer's inspection policy choice is independent of the supplier's wholesale price, w. The intuition is that the same wholesale price is incurred by the buyer regardless of the input and output quality of the product.

The above results clearly indicate that the threshold quality level is a function of both the inspection accuracy option and the cost of inspection. Therefore, in the rest of the analysis we denote this threshold value by

$q_t^B(i)$

where

$i=H,L$

distinguishes the results due to the selection of inspection accuracy. Before presenting the optimal inspection accuracy choice for the buyer, we define

${\gamma }_{HL}$

as follows:

$${\gamma }_{HL}=1-\frac{\beta ({\psi }_H-{\psi }_L)-\lambda ({\varphi }_H-{\varphi }_L)}{\beta g({\varphi }_H-{\varphi }_L)}.$$

13

Proposition 3

For given inspection technology choices (

${\varphi }_H,{\psi }_H$

) and (

${\varphi }_L,{\psi }_L$

),

• (i)

  if

  ${\psi }_H/{\varphi }_H\le {\psi }_L/{\varphi }_L$
  , then
  $q_t^B(H)\ge q_t^B(L)$
  . Consequently, the buyer chooses no inspection if
  $q\ge q_t^B(H)$
  and full inspection with option (
  ${\varphi }_H,{\psi }_H$
  ) otherwise. Inspection option with low accuracy is never a choice for the buyer in this case;

• (ii)

  if

  ${\psi }_H/{\varphi }_H>{\psi }_L/{\varphi }_L$
  , then
  $q_t^B(H)<q_t^B(L)$
  . Consequently, the buyer chooses no inspection if
  $q\ge q_t^B(L)$
  . When
  $q<q_t^B(L)$
  ,

Similar to the integrated system, the choice of inspection accuracy depends on the trade‐off between inspection costs and accuracy levels. The inspection option with lower marginal cost with respect to its accuracy is preferable to the buyer, as expected. In general, either one of the inspection accuracy options dominates the other or the buyer switches from low accuracy to high accuracy as the input quality decreases. Specifically, the high‐accuracy option always dominates the low‐accuracy option when the former one has lower cost to accuracy ratio. In this case, the buyer has no incentive to switch to the latter option due to inferior return on investment. In the opposite case, the comparison between high‐ and low‐accuracy options depends on another threshold given in (14), which is a function of price sensitivity of demand (β), impact of quality on demand (λ), external failure cost (g), and buyer's share on external failure cost (

$1-\gamma$

). We let

$q_t^B(HL)$

denote this threshold. In general, if the price sensitivity overweighs the impact of quality, the buyer will be reluctant to switch to high‐accuracy option as the input quality decreases. On the other hand, when the impact of quality on demand is significant, the buyer will be compelled to switch from low accuracy to higher accuracy as the input quality decreases, in order to stabilize demand and hence, the revenue. This is specially the case if the buyer's share in failure cost is high.

As an interesting final note for this stage, we observe that the impact of input quality on the resale price can be nonmonotone depending on whether the market demand is driven primarily by price or quality. It is straightforward to observe from (10) and (11) that the resale price decreases as input quality increases under both inspection regimes when

$\lambda <\beta (1-\gamma )g$

. When demand is not so sensitive to quality (i.e., λ is sufficiently low), any improvement in quality, and thus, reduction in external failure costs, results in increased demand through lower prices. In this case, quality exerts a “cost reducing” effect, where the gains obtained in external failure cost via increased quality enables the buyer to enhance her revenue by reduced resale price (Figure 2a). With sufficiently large λ, the “demand‐enhancing effect” overweighs the cost‐reducing effect of quality. As such, the equilibrium price increases with input quality in response to increased demand driven by higher quality (Figure 2b).

FIGURE 2

Input quality versus resale price

The supplier's input quality and wholesale price decisions

After incorporating the buyer's best response derived in the previous subsection, we can write the supplier's profit function under no inspection (

${\mathrm{\Pi }}_S^n$

) as the following:

16

Under full inspection, her profit function (

${\mathrm{\Pi }}_S^f$

) is

17

In this stage, inferring on the buyer's best response strategies on inspection and pricing, the supplier decides on her input quality followed by the wholesale price. To facilitate the analysis and follow the intuitive sequence of decisions, we first investigate the supplier's optimal wholesale price decisions for a given input quality level. It is straightforward to observe from the second derivatives, both functions given in (16) and (17) are concave in w and the first‐order optimality conditions yield the following optimal wholesale prices under no inspection (

$w_n$

) and full inspection with accuracy option i (

$w_f(i)$

):

$$w_n=\frac{\alpha +\lambda q+\beta (c+(1-q)v-(1-q)(1-2\gamma )g)}{2\beta }\text{and}$$

18

$$w_f(i)=w_n+{\varphi }_i\frac{(1-q)(\lambda +\beta (1-2\gamma )g-\beta {\psi }_i}{2\beta }.$$

19

Once we incorporate these values into the supplier profit functions, we can make the following first observation regarding input quality. Lemma 3

For

$s\le {(\lambda +\beta (v+g))}^2/(4\beta )$

, the supplier always chooses the perfect quality (i.e.,

$q=1$

) and no inspection is employed at equilibrium. Conversely, supplier's profit functions under all inspection regimes are concave in q, where optimal input quality is nonincreasing with s.

The above lemma indicates that when the impact of quality on demand (demand enhancing impact) or the external failure cost (cost reducing impact) is sufficiently high, the supplier has an incentive to build perfect quality, making inspection redundant. Noting that the supplier's quality decision is constrained by the buyer's threshold given in Proposition 2, we can generalize the above result and provide the following sufficient conditions on quality cost coefficient leading to the no‐inspection policy at equilibrium. Lemma 4

For a given inspection option with

${\psi }_i$

and

${\varphi }_i$

, there exists a unique threshold for quality cost coefficient,

$s_t(i)$

, below which the supplier chooses a unique input quality level,

$q_n$

, such that

$$q_n=\min (1,\max (q_t^B(i),q_n^{\ast })),$$

20

where

$$q_n^{\ast }=\frac{(\alpha -\beta (c+v+r))(\lambda +\beta (v+g))}{4\beta s-{(\lambda +\beta (v+g))}^2},$$

21

and there is no inspection at equilibrium.

The above result indicates that when the cost of quality is sufficiently low, the supplier will have incentive to build her input quality level above the buyer's threshold,

$q_t^B(i)$

and thus, the latter opts out of performing inspection. It is straightforward to observe from Proposition 2 that

$q_n^{\ast }\ge q_t^B(i)$

if and only if the supplier's external cost share is sufficiently high. That is,

$$\gamma \ge 1-\frac{\beta {\psi }_i-(1-q_n^{\ast }){\varphi }_i\lambda }{(1-q_n^{\ast }){\varphi }_i\beta g}.$$

22

We let

${\gamma }_n(i)$

denote the threshold value given in the right‐hand side of inequality (22). In this case, the supplier is actually better off with committing to higher shares in external failure costs in that higher shares relaxes the constraint due to the buyer's quality threshold. Clearly, there are two threshold cost coefficient values corresponding to the high‐ and low‐accuracy inspection options. We let

$s_t(H)$

and

$s_t(L)$

denote these threshold values, respectively. If the quality cost coefficient is below

$\min (s_t(H),s_t(L))$

, the supplier chooses an input quality level leading to no inspection at equilibrium. To analyze the case with high‐quality cost, we first introduce the supplier's unconstrained optimal quality choice under inspection option i:

$$q_f{(i)}^{\ast }=\frac{(\alpha +{\varphi }_i\lambda -\beta (c+{\psi }_i+v+(1-{\varphi }_i)g))((1-{\varphi }_i)\lambda +\beta (v+(1-{\varphi }_i)g))}{4\beta s-{((1-{\varphi }_i)\lambda +\beta (v+(1-{\varphi }_i)g))}^2}.$$

23

The above equation can be easily derived from the first‐order optimality condition applied to the supplier's profit function (17) evaluated at

$w=w_f$

. Next, we make the following observation. Lemma 5

For a given inspection option with

${\psi }_i$

and

${\varphi }_i$

,if

$s>s_t(i)$

, the supplier selects

$q_f(i)=\min (q_t^B(i),q_f^{\ast }(i))$

as her input quality resulting in full inspection. Moreover,

$q_f^{\ast }(i)\le q_t^B(i)$

if and only if

$$\gamma \le 1-\frac{{\psi }_i\beta -(1-q_f^{\ast }(i)){\varphi }_i\lambda }{(1-q_f^{\ast }(i)){\varphi }_i\beta g}.$$

24

We let

${\gamma }_f(i)$

denote the threshold value given in the right‐hand side of inequality (24). This inequality implies that when the supplier's (respectively, buyer's) share in external cost is sufficiently low (respectively, high), the supplier's unconstrained optimal quality choice will be below the buyer's quality threshold. The threshold

${\gamma }_f(i)$

decreases in

$q_f(i)$

and as such, increases in s. On the other hand, when γ is above this threshold, although the supplier may prefer a full‐inspection policy, her unconstrained quality choice will not be below the buyer's quality threshold. In this case, the supplier's quality constraint imposed by the buyer's threshold must be binding.

The cases given in Lemmas 4 and 5 capture the supplier's preference between no‐inspection and full‐inspection regimes. In addition to these cases, the supplier may need to choose between two available full‐inspection options signified by their costs and accuracy levels. It can be deduced from Proposition 3 that the switch between two full‐inspection options may occur only if

${\psi }_H/{\varphi }_H>{\psi }_L/{\varphi }_L$

. The following lemma lays out the equilibrium outcome in this case. Lemma 6

When

${\psi }_H/{\varphi }_H>{\psi }_L/{\varphi }_L$

and

$\beta ({\psi }_H-{\psi }_L)<({\varphi }_H-{\varphi }_L)(\lambda +\beta r)$

, given

$q_t^B(HL)$

as defined in Proposition 3, there exist two threshold values,

${\widehat{s}}_t(L)$

and

$s_t(HL)$

, where:

If

${\widehat{s}}_t(L)\ge s_t(HL)$

, then

• if

  $s\le s_t(HL)$
  , at equilibrium,
  $q=q_n$
  resulting in no inspection as detailed in Lemma 4;

• if

  $s>s_t(HL)$
  , at equilibrium,
  $q={\widehat{q}}_f(H)$
  resulting in full inspection with high accuracy, where
  ${\widehat{q}}_f(H)=\min (q_t^B(HL),q_f(H))$
  .

If

${\widehat{s}}_t(L)<s_t(HL)$

, then

• if

  $s\le {\widehat{s}}_t(L)$
  , at equilibrium
  $q=q_n$
  resulting in no inspection as detailed in Lemma 4;

• if

  ${\widehat{s}}_t(L)<s<s_t(HL)$
  , at equilibrium,
  $q={\widehat{q}}_f(L)$
  resulting in full inspection with low accuracy, where
  ${\widehat{q}}_f(L)=\max (q_t^B(HL),q_f(L))$
  ;

• if

  $s\ge s_t(HL)$
  , at equilibrium,
  $q={\widehat{q}}_f(H)$
  resulting in full inspection with low accuracy, where
  ${\widehat{q}}_f(H)=\min (q_t^B(HL),q_f(H))$
  .

Moreover,

${\widehat{s}}_t(L)\ge s_t(L)$

always holds.

We know from Proposition 3 that when

${\psi }_H/{\varphi }_H\le {\psi }_L/{\varphi }_L$

, the high‐accuracy option always dominates the low‐accuracy option because the former returns more bang for the buck. On the other hand, when the cost to accuracy ratio is too high for the high‐accuracy option, that is,

${\psi }_H/{\varphi }_H>{\psi }_L/{\varphi }_L$

and

$\beta ({\psi }_H-{\psi }_L)\ge ({\varphi }_H-{\varphi }_L)(\lambda +\beta g)$

, it is always dominated by the low‐accuracy option. The above lemma shows that when the gap between the two inspection options is sufficiently small in terms of cost and accuracy, the equilibrium inspection policy may move from low‐ to high‐accuracy option as the cost of quality increases. In this case, as the low‐accuracy inspection option includes the additional constraint stipulating that the input quality must be above

$q_t(HL)$

, the corresponding cost threshold,

${\widehat{s}}_t(L)$

, must be greater than or equal to

$s_t(L)$

, which ignores the high‐accuracy inspection option (when this option is dominated for the buyer). As such, the range for the quality cost resulting in no inspection effectively grows with respect to the case where the high‐accuracy inspection option is dominated for the buyer. The increase in the cost threshold in this case leads to an interesting result: the supplier may balk at choosing a quality level resulting in full inspection if a moderate‐cost‐higher‐inspection‐accuracy option is introduced by the buyer. More precisely, when s is in (

$s_t(L),\min ({\widehat{s}}_t(L),s_t(HL))$

), while the supplier's decision leads to full inspection with low accuracy in the absence of the high‐accuracy option, it results in higher input quality investment and thus, no inspection when the latter option is present. This counterintuitive result can be explained by the fact that under the supplier's quality investment tailored for low‐accuracy inspection, the buyer would prefer to adopt the high‐accuracy inspection once the latter option is made available. In this case, the subsequent decrease in the supplier's profit margin and increase in the external failure cost would not justify the full‐inspection regime for her. Consequently, she would be better off with increasing her input quality investment and avoiding full inspection.

Whether the inspection regime moves from no inspection to low‐accuracy inspection before switching to high‐accuracy inspection depends on how the parties share the external failure cost. In general,

${\widehat{s}}_t(L)$

decreases and converges to

$s_t(L)$

and

$s_t(HL)$

increases with γ. When

$s_t(HL)$

, or equivalently γ, is sufficiently high, low‐accuracy inspection becomes the preferable choice when the cost of quality is neither small nor large. In that case, as the cost increases, the equilibrium first shifts from no inspection to full inspection with low accuracy and then from low accuracy to high accuracy. On the other hand, when γ is low, in other words, when the buyer's share in external failure cost is high, she has more incentives to adopt the high‐accuracy inspection. This results in elimination of the low‐accuracy inspection as a possible equilibrium outcome. To further investigate the impact of γ on the equilibrium outcome, we define

${\overline{\gamma }}_f(i)$

as follows:

$${\overline{\gamma }}_f(i)=1-\frac{({\psi }_H-{\psi }_L)\beta -(1-q_f^{\ast }(i))({\varphi }_H-{\varphi }_L)\lambda }{(1-q_f^{\ast }(i))({\varphi }_H-{\varphi }_L)\beta g}.$$

25

Proposition 2 and Lemma 6 together imply that the unconstrained input quality under full inspection with high accuracy (i.e.,

$q=q_f^{\ast }(H)$

) can be achieved only if

$\gamma \le {\overline{\gamma }}_f(H)$

, whereas

$q=q_f^{\ast }(L)$

is possible only if

$\gamma \ge {\overline{\gamma }}_f(L)$

. Otherwise, the equilibrium input quality is

$q_t^B(HL)$

.

Finally, Lemmas 4–6 together lead to the following generalization for the supplier's equilibrium investment strategy on input quality. Proposition 4

For given inspection technology choices (

${\varphi }_H,{\psi }_H$

) and (

${\varphi }_L,{\psi }_L$

), Table 2 outlines the supplier's optimal input quality choices and the equilibrium inspection regimes.

TABLE 2

Input quality at equilibrium

Case I: ${\psi }_{\mathbf{H}}/{\varphi }_{\mathbf{H}}\le {\psi }_{\mathbf{L}}/{\varphi }_{\mathbf{L}}$
If $s\le s_t(H)$ then $q=q_n$ & $U=0$ (see Lemma 4)
If $s>s_t(H)$ then $q=q_f(H)$ & $U=1$ (see Lemma 5)
Case II: ${\psi }_{\mathbf{H}}/{\varphi }_{\mathbf{H}}>{\psi }_{\mathbf{L}}/{\varphi }_{\mathbf{L}}$ and $\beta ({\psi }_H-{\psi }_L)\ge ({\varphi }_H-{\varphi }_L)(\lambda +\beta (1-\gamma )g)$
If $s\le s_t(L)$ then $q=q_n$ & $U=0$ (see Lemma 4)
If $s>s_t(L)$ then $q=q_f(L)$ & $U=1$ (see Lemma 5)
Case III: ${\psi }_{\mathbf{H}}/{\varphi }_{\mathbf{H}}>{\psi }_{\mathbf{L}}/{\varphi }_{\mathbf{L}}$ and $\beta ({\psi }_H-{\psi }_L)<({\varphi }_H-{\varphi }_L)(\lambda +\beta (1-\gamma )g)$ (see Lemma 6)
${\widehat{s}}_t(L)<s_t(HL)$	${\widehat{s}}_t(L)\ge s_t(HL)$
If $s\le {\widehat{s}}_t(L)$ then $q=q_n$ & $U=0$	If $s\le s_t(HL)$ then $q=q_n$ & $U=0$
If ${\widehat{s}}_t(L)<s\le s_t(HL)$ & then $q={\widehat{q}}_f(L)$ & $U=1$	If $s>s_t(HL)$ then $q={\widehat{q}}_f(H)$ & $U=1$
If $s>s_t(HL)$ then $q={\widehat{q}}_f(H)$ & $U=1$

Cases I and II follow from Proposition 3 and Lemmas 4 and 5. Case III is a result of Proposition 3 and Lemma 6. Cases I–III are driven by the buyer's best response on inspection policy preferences, which are determined by the trade‐off between costs and accuracy levels of the inspection technology options. These cases are significant for the supplier in that they shape the quality threshold constraints in the decision model. In the first two cases, one of the inspection technology options dominates the other for the buyer (high accuracy in Case I and low accuracy in Case II). Consequently, the supplier cannot include the dominated option in her decision. In these cases, the equilibrium inspection policy depends only on the cost threshold corresponding to the dominating option.

The equilibrium outcome is relatively more ambiguous in Case III, where no‐inspection accuracy option dominates the other, and as such, both options can be adopted by the buyer depending on the input quality. As both options are potentially viable for the buyer, the supplier's cost thresholds are different under this case as detailed in Lemma 6. The change in input quality as a function of s depends on the relation between these thresholds. The equilibrium values for prices, quality levels, demand, and profits are summarized in Table A1 in Appendix A.

Interestingly, we observe that both input and end‐item quality, the resale price, demand, and player profits are independent of the supplier's external failure cost share (i.e., γ) when the quality threshold constraints in the supplier's decision model are nonbinding. This can be explained by the fact that in these cases, the buyer offsets external failure costs with her pricing decision. However, when γ is sufficiently small or sufficiently large, the buyer's quality threshold constraint on the supplier's decision model is activated. In such cases, as the quality thresholds are functions of γ, all outcomes are directly affected by its exact value of γ. Specifically, the supplier's input quality is nonincreasing with γ. At first glance, this is a counterintuitive result because one would expect that the supplier invests more if her cost share increases for the defect items. However, with the supplier's higher cost share, the buyer will be less inclined toward full inspection. Consequently, the buyer's quality threshold, and hence the supplier's input quality, decreases with γ when the quality threshold constraint is binding in the supplier's decision model.

As expected, the input quality and players' profits strictly decrease in the cost of quality. Although end‐item quality decreases in cost of quality under a given inspection regime, it may increase when the inspection regime switches from no inspection to full inspection (or from low‐accuracy inspection to high‐accuracy inspection). This is especially true when inspection accuracy is high, which would eliminate a significant portion of the nonconforming items. The impact of s on prices is relatively more ambiguous and depends on the trade‐off between the quality's impact on demand (i.e., λ) and failure costs (i.e., v and g). When λ is sufficiently small (i.e., the cost‐reducing impact of quality is more dominant), resale price increases in the cost of quality under a given inspection regime. Basically, the buyer is compelled to increase the price to counter balance the increasing external failure cost. On the other hand, when λ is large (i.e., the demand‐enhancing impact of quality is more dominant), the resale price decreases in cost of quality in order. In this case, the buyer is compelled to lower the price to balance the drop in demand due to lower quality.

In general, impact of quality on demand (λ) is an important factor for the inspection policy outcome. When λ is sufficiently large, the supplier's input quality choice will also be sufficiently high so that inspection is not needed. On the other hand, when λ is very small, inspection is not an equilibrium outcome as well. In this case, even if the input quality is low, inspection is not justified due to the high cost of inspection. Higher quality attained from inspection does not bring about higher sales volume because λ is small. The small increase in sales volume then does not justify the cost of inspection. When λ is neither too small nor too large, full inspection will be the equilibrium outcome. Consequently, we conclude that impact of λ on

$s_t$

is not monotonous as illustrated in Figure 3.

FIGURE 3

Threshold curves for input quality cost as function of λ

External failure cost share selection

In our discussion so far, the external failure cost share is assumed to be exogenous. The obtained results reveal that although equilibrium decisions and players' profits may be independent of γ (when the threshold quality constraint is nonbinding), the choice of inspection regime at equilibrium is driven by it because the buyer's threshold quality is a function of γ. As such, the value region for γ, more than its exact value, is critical for both players. In this subsection, we investigate the strategies of the players about the selection of γ. First, we focus on the case where the supplier makes this decision and later we discuss the buyer's best strategy. In all cases, the decision of the respective player on γ precedes all other decisions in the sequence of events.

It is straightforward to deduce from the analysis in the previous subsection that if the supplier decides on γ, she will prefer values that lead to unconstrained solutions for the input quality. Consequently, the quality cost thresholds would depend on the comparison of the unconstrained equilibrium profits. We can deduce from Lemma 4 that when cost of quality is sufficiently low (i.e., s is below the no‐inspection threshold), the supplier's selection for γ will be above

${\gamma }_n(i)$

resulting in no inspection with

$q=q_n^{\ast }$

. As mentioned earlier, in this case, the supplier is willing to absorb a higher share and reduce the need for inspection for the buyer. When one of the inspection accuracy options dominates the other and her quality cost coefficient is above the threshold, the supplier's choice for γ will be below

${\gamma }_f(i)$

, where i corresponds to the dominating option (Lemma 5). In this case, the supplier delegates higher share in the external failure cost to the buyer, who will be compelled to opt with full inspection to avoid large penalties due to defective products. In return, the supplier charges a lower wholesale price as indicated by (19). When both inspection accuracy options are competitive, if the supplier is better off with the low‐inspection accuracy option, her choice will be any value in (

${\overline{\gamma }}_f(L),{\gamma }_{HL})$

. In this case, γ is neither too low, which would lead to full inspection with high accuracy, or too high, which would lead to no inspection. As cost of quality increases beyond a threshold, the supplier will prefer high‐accuracy inspection over low accuracy and consequently, her selection for γ will be below

${\overline{\gamma }}_f(H)$

(Lemma 6). Therefore, when the supplier decides on the external failure cost share, in all cases, the equilibrium quality, resale price, and player profits are independent of the exact value of γ.

The case where the buyer decides on γ is more involved. A comparison of equilibrium profits reveals that the quality cost thresholds for inspection are higher for the buyer than that of the supplier. This indicates that the buyer is less inclined toward full inspection in comparison to the supplier. This is intuitive as the buyer avoids the cost of inspection while the supplier prefers to free ride on full inspection performed by the buyer. This observation implies that when the supplier is better off with no inspection, so is the buyer. On the other hand, when full inspection leads to higher equilibrium profits for the buyer, it does so for the supplier as well. Consequently, when the cost of quality is sufficiently low, the buyer's choice of γ would be greater than

${\gamma }_n(i)$

and when it is sufficiently large she would select a value below

${\gamma }_f(i)$

. In this respect, the incentives of both players are aligned in both cases. However, when the cost of quality (i.e., s) is neither too low nor too large, the buyer is better off with no inspection, while the supplier prefers full inspection. In this conflict zone, the buyer's choice of γ will be zero so as to provide incentive to the supplier to choose an input quality high enough to result in no inspection. When the two inspection accuracy options are both competitive, the preferences of the two players may conflict. In general, the buyer is better off with low‐accuracy inspection and small external cost share (i.e.,

$\gamma >{\gamma }_{HL}$

as per Proposition 3).

COMPARISON BETWEEN INTEGRATED AND DECENTRALIZED CHANNELS

In general, under both the integrated and decentralized settings, if the cost of quality is sufficiently low, investment in input quality will be high and inspection will not be employed, But, when quality cost is sufficiently high, both settings result in full inspection. However, the inspection policy choice is ambiguous when the quality cost is in between. Before investigating these choices across two structures in detail, we first compare the outcomes under a shared inspection policy. Proposition 5

Ceteris paribus (for a given inspection policy—full or no), the comparison of the two supply chain structures yields the following.

The integrated system chooses a higher input quality level than the supplier in the decentralized supply chain. That is,

$q_n^I\ge q_n$

under no inspection and

$q_f^I>q_f$

under full inspection.

Consequently, the end‐item quality of the integrated system is better than the decentralized supply chain, that is,

${\theta }_n^I\ge {\theta }_n$

under no inspection and

${\theta }_f^I\ge {\theta }_f$

under full inspection.

The integrated system charges a lower price to the end consumers implying that

$p_n^I\le p_n$

under no inspection and

$p_f^I\le p_f$

under full inspection.

The proof for the Proposition follows from the direct comparison of solutions provided in Tables 1 and A1. The above result implies that the integrated system can take advantage of its efficiency (no double marginalization) to provide a higher quality product at a lower price for customers, as long as the inspection choices are aligned in both settings. The extent to which the cost advantage is used to provide higher input quality and accuracy depends on how much the demand is sensitive to price and quality. Specifically, if demand is very sensitive to quality (i.e., large λ), the integrated system invests more in improving end‐item quality; if the demand is very sensitive to price, the investment is relatively less and investments are used primarily to reduce quality‐related costs. Obviously, the immediate question that arises then is whether the inspection choices are always aligned for the two supply chain structures. The following result shows that this is actually not the case. Lemma 7

Given any inspection technology option i,

$s_t(i)<s_t^I(i)$

always holds.

The above lemma implies that when

$s\le s_t(i)$

, both integrated and decentralized supply chains invest high in input quality and so they opt not to inspect any product; on the other hand, when

$s>s_t^I(i)$

, both of them do not invest much in input quality and counter it by opting for full inspection. However, for moderately high values, that is,

$s\in (s_t(i),s_t(i)]$

, the buyer in the decentralized supply chain opts for full inspection while the no inspection is the optimal choice in the integrated system. In this region, due to double marginalization, the supplier cannot justify investing in higher input quality and thus, the buyer is compelled to select full inspection at equilibrium. In contrast, the integrated firm's profit margin is large enough to generate sufficiently high input quality rendering inspection unnecessary. In that case, the decentralized supply chain may indeed produce a higher end‐item quality compared to the integrated system, if the inspection accuracy is sufficiently high. As full inspection is adopted by the decentralized supply chain in that region compared to no inspection in the integrated system, the post‐inspection quality in the decentralized supply chain may be higher, even though the integrated system invests more in input quality. The intuition is that although higher quality does not increase demand for the end‐item significantly, the players in the competitive case are compelled to increase end‐item quality via inspection so as to eschew external failure penalties. We provide an illustrative example in Figure 4 that shows that the decentralized end‐item quality θ is higher for medium s and that in both supply chains θ is nonmonotone in terms of s. Although, theoretically speaking, the price charged to the end consumers by the decentralized supply chain can also be lower in this case, our numerical experiments suggest that this is relatively rare; the double marginalization effect weighs in and the integrated system normally charges a lower price.

FIGURE 4

End‐item quality as function of quality cost under the integrated system and the decentralized supply chain

Under both channel structures, the low‐accuracy inspection option is always dominated by the high‐accuracy inspection option if the cost‐accuracy ratio for the former one is greater than that of the latter. Conversely, the preferences of the channels between inspection accuracy options may differ depending on γ. It can be deduced from Propositions 1 and 4 that when γ is lower than the ratio between internal and external failure costs, that is,

$v/(v+g)$

, there is a range for the differences between inspection costs, in which the full inspection with high‐accuracy option is preferable for the decentralized supply chain while it is completely dominated under the integrated system. In such cases, in addition to what is depicted in Figure 4, the decentralized supply chain may result in higher end‐item quality compared to the integrated system even under high‐quality cost. With small γ, the buyer's share of failure cost is relatively high, which compels her to adopt high‐accuracy option in order to generate high quality, while it is never a viable option for the integrated system.

CONCLUDING DISCUSSION

Quality and price are perhaps two of the most important factors in purchasing decisions of customers in a wide variety of sectors. Quality is especially interesting because it has both a revenue‐generating aspect as well as a cost‐reducing facet. The quality of the end product in many sectors is shaped by two elements—the quality of the input from the supplier and the inspection technology/policy of the buyer. Investments in input quality and inspection influence prices and hence, demand, through their impact on costs and quality. There is an interesting interaction between input quality, inspection policy and prices, and this interaction is also affected by the structure of the supply chain. In this paper, we study this interplay under two supply chain structures, namely, integrated and decentralized.

Our analysis first verifies that the preferred inspection policy is either no inspection or full inspection depending on the input quality for both supply chain structures and how they share the external failure cost for the decentralized supply chain. In our study, we observe two intriguing aspects of quality management for both types of chain structures. First, due to the cost reducing effect associated with higher conformance quality, end consumer prices do not necessarily increase with better quality. Second, higher cost of input quality does not necessarily result in lower end‐item quality due to the moderating effect of the inspection policy. Consequently, managers need to be careful about whether to focus on the cost‐reduction or the revenue‐enhancement feature of conformance quality. If the end customers are more sensitive to price, the cost‐reduction effect should be dominant; if they are more quality‐sensitive, the revenue‐enhancing feature should be the driver. Our analysis also partially explains why prices alone are not a good indicator of quality in consumer markets as has already been established empirically.

As expected, our analysis reveals that the impact of quality on demand and how the supply chain partners share the external failure cost are critical in shaping the inspection regime. However, the extent of their influence on the equilibrium is not entirely intuitive. Our results indicate that the relation between the impact of quality on demand and the inspection policy choice is not monotonic. Inspection is not a preferred option when the impact is either too small or too large. In the former case, additional investment on inspection does not justify the growth in demand and hence the revenues. In the latter case, the input quality is sufficiently high (i.e., close to perfect) and as such, inspection will not be needed. Interestingly, we observe that the impact of the external failure cost share is also somewhat ambiguous in that a channel partner might be better off with a higher external failure cost share in the decentralized supply chain. In general, the buyer is inclined toward employing inspection as her cost share increases. When cost of quality is below a threshold, the supplier is better off with a no‐inspection equilibrium. However, when the cost is not significantly below this threshold, the supplier's quality choice may still be below the buyer's quality threshold and thus, leading to full inspection if the cost share of the latter one is high. In such cases, the supplier would be better off with a lower share in cost for the buyer (respectively, higher share for herself), which ensures no inspection. On the other hand, when the cost of quality is sufficiently high, both parties are better off with full‐inspection equilibrium, which can be attained when the buyer's cost share is high. Moreover, when there are two competing inspection accuracy options, the buyer may be better off with a higher cost share to ensure the higher inspection accuracy outcome, if the cost of quality is prohibitive. We also observe that introducing an inspection technology option does not necessarily make full inspection more likely as an equilibrium strategy. In fact, introducing a new inspection option may create incentive for the supplier to revert back to choosing an input quality level resulting in no inspection. This is particularly the case when an inspection option with higher accuracy and moderate inspection cost is introduced. Basically, with the new option, the buyer's preference deviates from low‐accuracy to high‐accuracy option imposing additional constraint for the supplier's decision model under full inspection. This constraint degrades her profits under full inspection to the point that she is better off choosing an input quality level resulting in no inspection from the buyer.

Our comparison between the integrated and decentralized channels reveals that the decentralized supply chain is more likely to adopt the full‐inspection strategy compared to the integrated system. Specifically, when the investment cost for input quality is relatively low, both channels invest significantly in input quality and opts for no inspection, and when the investment cost is high, then they invest less and opt for full inspection. Due to the inefficiency created by the double marginalization effect in the decentralized supply channel, the supplier actually invests less in input quality. Interestingly, the low level of investment in input quality by the supplier means that the buyer is forced to go for full inspection in regions where the integrated system can afford no inspection. For low and high levels of investment cost for input quality, the integrated system delivers a better quality product because it has higher levels of input quality. It also charges a lower price, although the extent of benefits in terms of quality and price depends on how sensitive is the end customer demand on those two factors. But, for regions where the buyer in the decentralized channel opts for full inspection while the integrated system opts for no inspection, competition actually results in higher end‐item quality (although the price might still be higher). We also demonstrate that this is likely to happen when the input quality is more expensive and inspection accuracy is cheaper. The rationale is that in those cases the supplier invests less in input quality forcing the buyer to select full inspection even earlier as the cost of quality increases compared to the integrated system. Moreover, if this inspection is highly accurate, then the decentralized system ends up with a better quality end product. So, indeed, in the decentralized setting, the interaction between input quality and inspection policy is actually exacerbated by incentive asymmetry.

The paper can be extended in the following three directions. First, we assumed a perfect information setting. An interesting extension concerns the case of unobservable supplier's quality by the buyer. A second extension is to study the case where the two parties in a decentralized system enter into an alliance and determine their quality investment decisions simultaneously. Finally, while we focus on vertical competition, it would also be worthwhile to study the impact of horizontal competition (i.e., in an oligopolistic market) on equilibrium quality management decisions.

Footnotes

1

All technical proofs are provided in Appendix .

ORCID iD

Haresh Gurnani

TABLE OF THE EQUILIBRIUM RESULTS “QUALITY INVESTMENT,INSPECTION POLICY,AND PRICING DECISIONS IN A DECENTRALIZED SUPPLY CHAIN”

PROOFS “QUALITY INVESTMENT,INSPECTION POLICY,AND PRICING DECISIONS IN A DECENTRALIZED SUPPLY CHAIN”