A CommitmentBased Explanation for Outsourcing Multiple Tasks

A Commitment-Based Explanation for Outsourcing Multiple Tasks* PEI-CHENG LIAO, National Taiwan University SURESH RADHAKRISHNAN, University of Texas at Dallas 1. Introduction Over the last two decades, there has been a considerable increase in outsourcing of products and services in the manufacturing and information technology (IT) arenas with the objective of reducing costs and improving efficiency and quality (see Willcocks, Lacity, and Fitzgerald 1995; Earl 1996; Bryce and Useem 1998; Costa 2001; Kakabadse and Kakabadse 2005; Weeks and Feeny 2008).1 A few observations on the trend in outsourcing practices are noteworthy. In the early years buyers and clients outsourced production-related tasks and induced vendors to improve production technology or quality (see Minahan 1998b; Y. Hwang, Rangtusanatham, and Pei 2006; Shi 2007; Y. Hwang, Erkens, and Evans 2009). In recent years clients have increased the number of tasks outsourced by including design-related tasks as well (see Minahan 1998a; Carbone 2000; Natovich 2003; Kakabadse and Kakabadse 2005).2 However, the increase in the price has not been commensurate with the direct costs of the additional tasks (see Earl 1996; Minahan 1998b; Milligan 2000). While the trend in increased tasks being outsourced is rationalized by the improved ability of vendors (Minahan 1998b; Shi 2007), the trend in prices not increasing to cover the direct costs of additional tasks is rationalized by the higher bargaining power of clients (see Minahan 1998b; Dyer 1996; Crook and Combs 2007). Our objective is to provide an alternative rationale based on agency issues for the outsourcing trend of increased tasks without a corresponding increase in price. We consider an agency problem with one client and one vendor. There are two actions: the design action and the production action. The client can observe the vendor’s production action, but it is not contractible. If the client observes the vendor exerting an undesirable production action, then the client can help the vendor out with an intervention action.3 Knowing *

1.

2.

3.

Accepted by Raffi Indjejikian. We gratefully acknowledge insightful comments from Raffi Indjejikian and two anonymous referees. Pei-Cheng Liao also acknowledges financial support from the College of Management, National Taiwan University, and National Science Council of Taiwan under Grants No. NSC 982410-H-002-039 and 99-2918-I-002-010. A survey conducted by Manpower UK Ltd. (http://www.manpower.com/) in 2000 shows that 68% of UK companies outsource some of their activities. Among the companies that outsource, 17 percent outsource IT activities and 15 percent outsource manufacturing activities. Lacity and Willcocks (2000) state that the global IT outsourcing market revenues grew from US$9 billion in 1990 to US$121 billion in 2001 (also see http://www.gartner.com/). In the IT industry, vendors take on more responsibility in the requirement planning and design stages. In the electronics industry, full service vendors take on the responsibilities of design, development, prototype creation, and production (see Minahan 1998a). Examples of the vendor expecting to be helped out by the client abound in casual conversations with purchase and IT project managers. Ho, Ang, and Straub (2003: 73–74) state the viewpoint of a client as follows: “I have to review his work, roles, and responsibilities thoroughly…. There’s no value added by the contractor organization, and I don’t see the service level growing.” Shi (2007) points out that the vendor’s pace of technology change may be too slow to meet the client’s real needs, resulting in the client stepping in to assist in this process. Earl (1996) points out that in IT service businesses, the client’s staff routinely provides support to the vendor, even though the vendor is the most established one in the IT service business. In a similar vein, iSuppli analyst Adam Pick says, “Many tech giants have tried to outsource manufacturing to companies in Asia, only to end up repeatedly sending teams of designers and engineers to help those companies get up to speed” (Hall 2009: 2).

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this, the vendor can choose to shirk and let the client help him out. Hence, the client has to ensure that the price is such that the vendor does not expect the client to help him out by exerting an intervention action, that is, the client’s commitment problem. We find that the client benefits from helping the vendor if the production technology is sufficiently good or effective. When the production technology is sufficiently good, the client’s opportunity loss from not intervening is high if the vendor shirks; thus, the client’s commitment problem is more severe than the vendor’s moral hazard problem. If the price simply induces the vendor to choose a desirable production action, then the vendor, knowing that the client will intervene if he shirks, will find it beneficial to shirk. To make sure that this does not occur, the client increases the price and gives a larger share of his profits to the vendor such that intervention is not attractive; this extra payment to the vendor is like a bribe to ensure that the vendor does not shirk.4 We show that the client’s profit increases as the production technology improves, up to the point when the vendor’s moral hazard problem ceases to be more severe than the client’s commitment problem. After that point, when the client’s commitment problem is more severe than the vendor’s moral hazard problem, improvements in production technology decrease the client’s profits: the extra profit due to improvement in technology is captured as rents by the vendor because of the commitment problem. We then examine whether outsourcing an additional task such as the design action, which is not subject to moral hazard, can help mitigate the client’s commitment problem.5 We consider a case where the direct cost of the outsourced design action is greater than the direct cost of the in-house design action. Thus, a conventional cost-benefit analysis would suggest that the design action should not be outsourced. However, if outsourcing the design action lowers the effectiveness of the client’s intervention action, then the client will not lose as much by not intervening when the vendor shirks and the severity of the client’s commitment problem is reduced as well.6 We find that whenever the client’s commitment problem is sufficiently severe in the in-house design case, outsourcing the design action is beneficial for the client because it helps to mitigate the client’s commitment problem. We show that by outsourcing the design action the price need not be as high as in the case where the design action is not outsourced. In effect, outsourcing the design action helps the client “cut back” the vendor’s rents. More importantly, this phenomenon does not arise simply because the bargaining power of the client is higher than that of the vendor, as suggested by the prior literature (Minahan 1998b). We thus provide an alternative rationale for the trend that even though the vendor’s responsibilities increase, the client’s price does not cover the costs of such increased responsibilities. 4.

5.

6.

Arya, Glover, and Sunder (1998) and Arya, Glover, and Radhakrishnan (2006) examine a similar commitment problem where the principal has to commit not to intervene. They do not examine the delegation of multiple tasks in the context of a make-or-buy decision. In a manufacturing context the design action involves creating part-drawings and setting up the production process parameters. In an IT context the design action involves requirements assessments and planning (see Jacob, Jayant, and Radhakrishnan 2011). In reality, while the design action is also subject to moral hazard, it is likely to be less severe than the moral hazard problem of a production action because performance benchmarks, that is, the drawings and the requirements, are jointly observable. We model this notion of the design action being subject to less moral hazard than the production action by assuming that the design action is not subject to moral hazard in order to focus on providing insights into outsourcing such actions. Examples of cases where the client is not able to help the vendor out in the production process, because he has delegated many important tasks such as design, abound in the vendor management literature (see, e.g., Barthelemy 2003; Goolsby 2006a,b; Shi 2007; Tadelis 2007; Weeks and Feeny 2008). These papers of course tout these aspects as outsourcing failures. Our notion here is that if the client is not engaged in the design, he loses expertise in the production as well. A good analogy is in research co-authorships. As senior faculty delegate more tasks to their junior colleagues, it may become more difficult for them to intervene and bail the junior co-authors out if there are problems in the research.

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Collectively, in settings characterized by commitment issues of nonintervention where outsourcing more tasks mitigates the commitment problem in the sense that outsourcing the additional tasks lowers the effectiveness of intervention of earlier tasks, the results suggest the following: (a) production improvements and quality improvements will have preceded outsourcing of additional tasks, and (b) the increase in unit price is not likely to cover the increase in cost of the additional tasks. Thus, broadly speaking, the increased trend in outsourcing may be due not only to vendors’ improved abilities and clients’ increased bargaining power, but also to the changing nature of agency issues.7 While the intuition for our results stems from Arya et al. 1998 and Arya et al. 2006, our insights extend Saouma 2008, who examines a client’s decision on whether to outsource two tasks instead of one task. Saouma (2008) examines the trade-offs between the double moral hazard problem when the client outsources one task and the moral hazard problem with two tasks when the assembly task is outsourced.8 Saouma (2008) shows that the client favors a moral hazard problem with two tasks rather than a double moral hazard problem because the same incentive used to motivate one task can be enough to motivate the assembly task: the moral hazard problem for one of the tasks can be resolved for “free.” We extend this insight into a setting where the additional task that is outsourced also helps to mitigate the client’s commitment problem. We thus provide an additional rationale for outsourcing more responsibilities/tasks to the vendor.9 Demski and Sappington (1993) examine a double moral hazard problem where the client’s private information is to be conveyed to the vendor and the vendor needs to be motivated to exert high effort. They show the existence of settings where the client’s private information is not used for contracting purposes. They demonstrate the subtleties involved in the make-or-buy decision when organization and information factors are included in the analysis. In this study, we examine another factor that could lead to friction — the inability of the client to precommit to not helping the vendor out. Interestingly, in contrast to Demski and Sappington 1993, we find that even though the delegation setting does not directly provide information on the vendor’s production action, it is used in contracting because it helps alleviate the precommitment problem. Our result is similar in spirit to the intuition in Arya et al. 2006: Informativeness is not necessary for a signal to be included in the contract when commitment issues similar to our model are considered. The rest of the paper is organized as follows: Section 2 lays out the model, Section 3 presents the analysis of the model and our results, Section 4 discusses the empirical implications and provides some concluding remarks. 2. Model Some trends in outsourcing The Taiwanese notebook industry provides a good example of trends in outsourcing. Leading U.S. and Japanese brands have increased the number of outsourced tasks and responsibilities from production and assembly in the 1990s to design, development, 7.

8.

9.

This evolutionary rationale provides an alternative explanation to the argument that vendors have become more capable and thus can handle more tasks. In our model, this argument would be tantamount to saying that the design action could not have been outsourced in the initial years, but it has only been possible to do so in recent years. We discuss other empirical predictions in the concluding section. The double moral hazard model has been used to address different issues in supply chain contracting literature (see, e.g., Cooper and Ross 1985; Demski and Sappington 1991; Bhattacharyya and Lafontaine 1995; Baiman, Fischer, and Rajan 2000; Balachandran and Radhakrishnan 2005; Corbett, DeCroix, and Ha 2005; Chao, Iravani, and Savaskan 2009). Other studies have examined various benefits of outsourcing. For example, Arya, Mittendorf, and Sappington (2008) examine the effects of competition among clients as a motivation to outsource. Arya and Mittendorf (2010) show how “loss leaders” in the final market will affect the price in the input market; see also Arya and Mittendorf 2007.

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production, distribution, after-sales service, and even logistics management in some cases in more recent years.10 Over the same time period the vendor’s production quality has improved considerably (see Engardio and Einhorn 2005; Einhorn 2005). However, the profit margins of the vendors have decreased; for example, the profit margin of Quanta Computer Inc., the world’s largest maker of notebook PCs and a key supplier to Dell and Hewlett-Packard, has shrunk from 18.79 percent in 1998 to 2.62 percent in 2010 (http:// www.quantatw.com/).11 Similar trends that characterize other electronics and IT companies may be summarized as follows: • In the early years clients outsourced production-related tasks and vendors improved their production-related quality both in terms of yield rates and final product quality (see Minahan 1998b; Y. Hwang et al. 2006; Shi 2007; Y. Hwang et al. 2009). • In recent years clients have increased the number of tasks outsourced by including design-related tasks as well (see Minahan 1998a; Carbone 2000; Natovich 2003; Kakabadse and Kakabadse 2005). However, the increase in the price has not been commensurate with the increase in the cost of these additional tasks, thus depressing the vendor’s profit margins (see Earl 1996; Minahan 1998b; Milligan 2000). The improved ability of vendors is typically provided as the rationale for increasing the number of tasks outsourced (Minahan 1998b; Shi 2007), and higher client bargaining power is provided as the rationale for the shrinking profit margins (see Minahan 1998b; Dyer 1996; Crook and Combs 2007). In this study, our objective is to provide an alternative rationale based on agency issues for the outsourcing trend of increased tasks without a corresponding increase in price to cover the direct costs of the additional tasks. The principal-agent model We consider a one-period supply chain with a risk-neutral client and a risk-neutral vendor. The client requires n units of a product, where without loss of generality we set n = 1. The product quality can be either good or bad and the probability that the unit is good is given by q and, correspondingly, the probability that the unit is bad is given by (1 – q). The quality of the product, q depends on a production action by the vendor, a; a potential intervention action by the client, b; and a design action exerted by either the client, DC, or the vendor, DV, that is, q(a, b, Dk) for k = C, V. As such, the probability of a good unit is given by q(ai, bj, Dk) and is denoted qijk for i, j = H, L and k = C, V. A good-quality product provides a benefit to the client of x > 0 per unit, and a bad-quality unit provides a benefit of x = 0. The vendor’s production action can be either high or low, that is, a ∈ (aH, aL) with a corresponding cost of aH > aL = 0. Similarly, the client’s intervention action can be either high or low, that is, b ∈ (bH, bL) with a corresponding cost of bH > bL = 0. We let the cost of the vendor’s production action be lower than the cost of the client’s intervention action, that is, bH > aH. We also let the vendor’s design cost be higher than the client’s design cost, that is, DV > DC. These assumptions on costs suggest that outsourcing the production action and performing the design in-house would be beneficial for the client. The vendor’s production action is observable to the client and the client’s intervention action is observable to the vendor. However, the production and intervention actions are not verifiable by a third party and hence are not contractible. In essence, the client can 10.

11.

The tasks/responsibilities of Taiwanese vendors increased from 39.2 percent in 1998 to 92.4 percent in 2008. The data for this example are from Market Intelligence & Consulting Institute (http://mic.iii.org. tw/) and Industry & Technology Intelligence Services (http://www.itis.org.tw/). So much so that Quanta Computer Inc. and two other leading notebook makers considered the threat strategy of declining HP’s orders. Casual conversations with sales executives confirm this trend. Similarly, casual conversations with purchase managers of buyers during their visits to vendors’ facilities confirm that they have to “hold the hand” in the production set-up phases.

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observe if the vendor chooses a low production action (aL), but cannot use this information to impose penalties on the vendor because this action cannot be verified and enforced in a court of law. This is one notion of incomplete contracts. However, if the client observes that the vendor chooses the low production action, he can choose the high intervention action (bH). The intervention action is similar to the production action and represents an action that makes good the vendor’s low production action. Examples of intervention actions include the client helping the vendor set up the production process when the client observes that the vendor is not doing an effective job, that is, exerting the low production action. The intervention action captures the essence of Goolsby’s 2003 characterization of the problem where the vendor expects the client to help him out: “All outsourcing arrangements, unfortunately, aren’t working optimally among all vendors, nor are the providers proactive in effectively resolving issues. Many experience a ‘vendor running to the buyer’ situation where vendors, unable to resolve issues ‘run’ to the buyer to make the business decision.” When the vendor does what is required of him, that is, exerts a high production action, the client does not intervene, that is, b = bL. The client pays the vendor s (zero) for each good (bad) unit.12 This payment scheme is similar to the client performing a perfect incoming inspection that classifies the good and bad units precisely (see Balachandran and Radhakrishnan 2005 and I. Hwang, Radhakrishnan, and Su 2006, who consider similar contracts).13 In the IT development setting, the good unit is the proof of concept or criteria on service-level agreements that are met (Jacob et al. 2011). Note that because the outcome of the inspection, that is, good or bad unit, is contractible, when a unit is identified as a good unit the client will have to pay the vendor even if the vendor had exerted the low production action and the client had intervened. The problem for the client is to design a contract such that the vendor is induced to exert the high production action without the client exerting the intervention action. In essence, the problem for the client is to design a unit price s such that he commits not to intervene. Representing the agency problem The sequence of events unfolds as follows. First, the client makes the sourcing decision on the design action and offers the vendor a contract. Second, based on the contract terms, the vendor chooses the production action. Third, the client observes the vendor’s production action and chooses whether to intervene. Last, the production takes place, and if the unit is good the vendor is paid s as specified in the contract. The expected profits for the client (Wijk) and the vendor (Uijk) are represented below, where the first subscript i denotes the vendor’s production action that is induced, the second subscript j denotes the client’s intervention action that is induced, and the last subscript k denotes whether the design action is exerted by the client or the vendor. Wijk ¼ ðx  sk Þqijk  bj  Dk ZC

ð1Þ;

Uijk ¼ sk qijk  ai  Dk ZV

ð2Þ;

12. 13.

It is optimal to set the unit price for the bad unit to be zero when the payments are restricted to being nonnegative. The insights with respect to the commitment problem will be qualitatively similar if we consider an imperfect incoming inspection in which the vendor is paid only when the inspection classifies the unit as good.

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where sk denotes the price for a good unit when the design action is exerted by k = C, V; ZC = 1 and ZV = 0 when k = C; and ZC = 0 and ZV = 1 when k = V. (1) is the expected net profit for the client, where the first term is the expected benefit after payment to the vendor for the good unit minus the last two terms, which are the costs of intervention and design actions respectively. (2) is the expected profit for the vendor and is given by the unit price for the good unit minus the costs of production and design actions respectively. We examine settings where the client induces the vendor’s high production action (aH), and chooses a low intervention action (bL).14 The problem for the client for each k = C, V is represented in the following program. MaxSk WHLK

Subject to UHLK  0

ðOBJÞ

ðPCÞ

UHLK  ULLK

ðICVÞ

WHLK  WLHK

ðICC1Þ

WLLK  WLHK

ðICCÞ

The objective function (OBJ) of the program is the client’s expected profit. Constraint (PC) is the participation constraint, which requires that the unit price provide the vendor with an expected profit that covers his reservation profit of zero. Constraint (ICV) is the incentive compatibility constraint of the vendor, which ensures that the unit price induces the vendor to choose aH given the unit price (sk) and the client chooses the low intervention action (bL). This is the standard incentive constraint in principal-agent models that ensure self-selection of the desired action. Constraint (ICC1) is the typical incentive compatibility constraint of the client’s commitment problem (for example, see Arya, Glover, and Sivaramakrishnan 1997). The unit price is chosen such that the client is indifferent between the required combination of the vendor’s high production action and his low intervention action, and the vendor’s low production action and his high intervention action. Constraint (ICC) is the incentive compatibility constraint of the client, which ensures that the unit price acts as a commitment device that signals to the vendor that the client will not intervene if the vendor exerts a low production action (aL). Essentially, constraint (ICC) ensures that the client is indifferent between repairing and not repairing the faulty work of the vendor, if the vendor chooses the low production action; that is, the client commits to not intervening and lives with the resulting loss due to bad quality. This constraint is similar to the incentive constraint in the subjective performance evaluation literature (see Bull 1987; Levin 2003). When the client/principal uses nonverifiable, subjective performance measures, he needs to commit to not misrepresenting to avoid paying 14.

We assume that benefit x is sufficiently large, such that the expected profit for the client is positive. It can be verified that if qHLC is sufficiently larger than qLLC; and qHLC = qLHC using assumptions bH > aH and x is sufficiently large, inducing aH, bL is optimal.

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the bonus — the commitment constraint makes the nonverifiable measure contractible. In a similar fashion in our setting, constraint (ICC) makes the vendor’s nonverifiable action contractible.15 Constraints (ICC1) and (ICC) represent the client’s commitment problem. To illustrate the significance of considering constraints (ICC1) and (ICC), consider the following numerical example for k = C. Let qHLC = 0.90, qLHC = 0.99, qLLC = 0.60, bH = 7, aH = 6, DC = 0, and x = 100. The solution to the program with only constraints (PC) and (ICV) is SC = 20: it can be verified that (ICV) is binding and (PC) is not binding. The expected profits for the vendor are: UHLC = (20 3 0.90) – 6 = 12, and ULHC = (20 3 0.99) – 0 = 19.80. Thus, the vendor is better off choosing the low action if the client will choose the high intervention action. The client’s expected profits are WHLC = [(100 – 20) 3 0.90] – 0 – 0 = 72, and WLHC = [(100–20) 3 0.99] – 7– 0 = 72.20. Thus, WLHC > WHLC; knowing this, the vendor will choose the low production action. Constraint (ICC1), the typical client’s commitment constraint, is not satisfied. Even after considering constraint (ICC1) as in the typical commitment problem, in our setting it is important to consider constraint (ICC) as well. Consider the program with (PC), (ICV) and (ICC1). It can be verified that constraint (ICC1) is binding and sC = 22.22. The expected profits for the vendor are UHLC = (22.22 3 0.90) – 6 = 14, and ULHC = (22.22 3 0.99) – 0 = 22.00. Thus, the vendor is better off choosing the low action if the client will choose the high intervention action. The client’s expected profits are: WHLC [(100 – 22.22) 3 0.90] – 0 – 0 = 70, WLHC = [(100 – 22.22) 3 0.99] – 7 – 0 = 70 and WLLC = [(100 – 22.22) 3 0.60] – 0 – 0 = 46.67. Thus, WLHC > WLLC; that is, the expected profit for the client when he observes that the vendor has chosen the low production action is higher when he intervenes than when he does not. Because ULHC > UHLC, the vendor will find it beneficial to choose the low production action, knowing very well that the client will intervene. Thus, the solution to the program with (PC), (ICV), and (ICC1) is not an equilibrium, in the sense that it cannot implement/sustain {aH, bL}. The main difference between the program and the typical principal-agent commitment models is the additional incentive compatibility constraint with respect to the commitment of nonintervention. The additional constraint is required here because the client does not have any other mechanism to precommit to a particular intervention action.16 Discussion of assumptions and modeling choices The design action is assumed to be observable and contractible.17 We make some intuitive simplifying assumptions with respect to the probability of the good unit. We make the standard assumption made in principal-agent models that the vendor’s high production action results in a higher probability of a good unit, that is, qHLK > qLLK for k = C, V; and similarly, the client’s intervention action results in a higher probability of a good unit, that is, qLHK > qLLK for k = C, V. We assume that qHLC = qHLV and qLLC = qLLV: given 15. 16.

17.

We would like to thank an anonymous reviewer for this suggestion. We do not imply that other mechanisms such as reputation and long-term relationships cannot mitigate the commitment problem. Other mechanisms to mitigate such commitment problems may not be implementable because identifying and fixing blame for failures may be difficult if not impossible (see Balachandran and Radhakrishnan 2005). Also, some projects, such as IT development projects, may not involve long-term relationships. Our analysis and representation help to provide insights into how the severity of the commitment problem affects outsourcing decisions and thus provide one explanation for the trend in outsourcing which can be sustained in equilibrium. We do not imply that the design action is not subject to moral hazard problems. Introducing noncontractibility of the design action will add another layer of agency problem but will not affect the insight obtained from the client’s commitment problem. Simply put, if the moral hazard problem with respect to the design action is relatively less severe than the moral hazard problem with respect to the production action, all of the results will continue to exist.

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that the client does not intervene, the probability of a good unit depends on the vendor’s production action only. This assumption implies that outsourcing the design action does not affect the effectiveness of the vendor’s production action, and helps to isolate the effect of the commitment problem from production effects. We further assume that qHHC = qHLC = qLHC, implying that when the design action is exerted by the client, the vendor’s production action and the client’s intervention action are perfect substitutes, in the sense that either one is enough. These assumptions are made to isolate the effects of intervention in settings where the vendor does not exert the high production action. In other words, under these assumptions the client will not want to induce the vendor’s high production action as well as intervene because doing so will not provide the client with any additional benefit. Correspondingly, when the design action is performed by the vendor, we assume that qHHV = qHLV. While these assumptions may be violated in specific settings, these assumptions are made to highlight the driving force of the commitment problem and thus are simply an experimental choice. Even when these assumptions are violated, the force of the commitment problem is important. In other words, the assumptions other than the one to be discussed next, are experimental choices made especially to highlight the effect of outsourcing the design action for reasons other than enhancing the productivity of the vendor’s production action. The important assumption for our result of outsourcing the design action to mitigate the commitment problem is that qLHC > qLHV, that is, the probability of a good unit is higher when the design action is exerted by the client than by the vendor. Intuitively, the assumption implies that when the design action is outsourced, the client loses the knowledge or ability to effectively intervene. That is, outsourcing the design action adversely affects the effectiveness of the client’s intervention action.18 This is reasonable in many circumstances: For example, many of the outsourcing failures point to “excessive” outsourcing in the sense that design-related tasks outsourced by the client lead to his losing the ability to help the vendor with production-related problems (see Goolsby 2003). On a similar vein, in the Ford-Firestone case, Ford was alleged to have overly delegated tasks and responsibilities (see Balachandran and Radhakrishnan 2005). We proceed to characterize the solution to the program for each k = C, V. 3. Analysis We now proceed to characterize the optimum solution to the program under each regime, that is, k = C, V where k = C denotes the design action done by the client and k = V denotes the design action done by the vendor. After characterizing the solution, we examine how the unit price and expected profits for the vendor change when qHLK for k = C, V increases, because this represents one characteristic of the outsourcing trends. We then derive conditions under which k = V yields the client a higher profit than k = C. No outsourcing of design action In this case, the client exerts the design action. The solution to the program is characterized in the following proposition.19 18.

19.

The spillover effect of this action to the effectiveness of the intervention/production action is the reason we refer to this action as the design action. Furthermore, as will be seen later in the analysis this assumption will directly help to mitigate the commitment problem as in Arya et al. 2006. Conversations with buyers in the notebook and IT industries point to buyers responding to vendors’ requests to help them with production-related issues with the reply that they do not know the details because they have outsourced the design action, and hence cannot help the vendors much. While this assumption directly drives our main insight, we provide a consistent argument for how outsourcing additional tasks can end up being optimal when quality improvements occur. Proofs of propositions are provided in the appendix.

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PROPOSITION 1. When the design action is performed by the client, the solution to the program is characterized as follows. (a) If the effectiveness of the production action is sufficiently large, then the client’s commitment problem is severe and the vendor receives an expected payoff greater than his reservation profit. Technically, if (T1) is satisfied, then the unit price, sC1, is determined by the (ICC) constraint, the vendor’s expected profit, UC1 is greater than the reservation profit of zero. The solution is given below. sC1 ¼ x  ½bH =ðqLHC  qLLC Þ; WC1 ¼ ½bH qHLC =ðqLHC  qLLC Þ  DC ; UC1 ¼ fx  ½bH =ðqLHC  qLLC ÞgqHLC  aH [ 0; x  xT1 ¼ ½bH =ðqLHC  qLLC Þ þ ½aH =ðqHLC  qLLC Þ:

ðT1Þ:

(b) If the effectiveness of the production action is not sufficiently large, then the client’s commitment problem is not severe and the vendor receives an expected payoff greater than his reservation profit. Technically, if (T1) is not satisfied, then the unit price, sC2, is determined by the (ICV) constraint and the vendor’s expected profit, UC2 is greater than the reservation profit of zero. The solution is given below. sC2 ¼ ½aH =ðqHLC  qLLC Þ; WC2 ¼ fx  ½aH =ðqHLC  qLLC ÞgqHLC  DC ; UC2 ¼ ½aH qLLC =ðqHLC  qLLC Þ [ 0:

Before discussing the results of Proposition 1, we discuss condition T1 and the threshold, xT1 . Condition T1 compares the benefit from the good unit to a threshold value, xT1 , which is made up of the cost-to-productivity ratios of the client’s intervention and vendor’s production actions. If the vendor’s cost-to-productivity ratio, [aH/(qHLC – qLLC)], is small (large) then the vendor’s moral hazard problem is less (more) severe, because the vendor will need less (more) incentives to induce the high production action. Similarly, for a given benefit from the good unit, x, if the client’s cost-to-productivity ratio, [bH/(qLHC – qLLC)], is small (large) then the client’s commitment problem is more (less) severe because the client will (not) find it beneficial to intervene when he observes that the vendor is shirking. Thus, condition T1 represents the severity of the client’s and vendor’s commitment problems; if the client’s commitment problem is more severe, then condition T1 is satisfied, and vice versa. As such, we refer to the case where condition T1 is satisfied, as the client’s commitment problem is severe. Proposition 1.1 characterizes the solution to the program when the client’s commitment problem is severe. Intuitively, when the client’s cost-to-productivity ratio is small, then the client’s profit can also be large with the client’s intervention action. The vendor knows that the client has more to lose by not intervening when the client observes the vendor’s low production action. Accordingly, the vendor will be more inclined to shirk when the client’s cost-to-productivity ratio is small (i.e., effectiveness of the client’s production action is large). This is consistent with popular press articles talking about the “vendor rushing to the client” in settings where the effectiveness of the production technology is sufficiently large such as in the IT and electronic industries (see Goolsby 2003). The client’s response to prevent the vendor’s shirking behavior and induce the vendor’s high production action is to pay him more, so as to decrease his own share and thereby make it unattractive for him to intervene. Specifically, the client pays the benefit he

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receives from the good unit less his cost-to-productivity ratio (sC1); this ensures that the client will not find it beneficial to intervene when he observes the vendor’s low effort.20 When the effectiveness of the production action is small, the solution is the same as the one obtained in classic principal-agent models with moral hazard problems and is given in Proposition 1.2. Effect of improvements in production technology, qHLC and qLHC We consider the impact of improvements in production technology because this is one of the characteristics of the trends in outsourcing: over time the trend has been toward improved yields of good units. We represent improvements in production technology as follows. Starting from qHLC ¼ qLHC ¼ q, we let the quality improvements be a nonnegative quantity denoted e  0, with qHLC ¼ qLHC ¼ q þ e representing recent years with improved quality. We assume that when e = 0, the client’s commitment problem is not severe, that is, Proposition 1.2 applies. The following proposition characterizes the effect of improvements in production technology. PROPOSITION 2. When the design action is performed by the client, qHLC ¼ qLHC ¼ q þ e with e  0 and at e = 0, the vendor’s moral hazard problem is more severe than the client’s commitment problem, that is, condition T1 is not satisfied; as e increases, ceteris paribus, the solution changes in the following manner. (a) The domain over which the client’s commitment problem is severe, that is, condition T1 is satisfied, increases: technically, ½d xT1 =de\0 . (b) The unit price and vendor’s expected profit decreases until condition T1 is satisfied and then increases, and the client’s expected profit increases until condition T1 is satisfied and then decreases, that is, [dsC2/de] < 0, [dUC2/de] < 0, [dWC2/ dε] > 0, when (T1) is not satisfied and [dsC1/de] > 0, [dUC1/de] > 0, [dWC1/ dε] < 0, when (T1) is satisfied. Proposition 2 shows the effect of improvements in production technology starting from a setting where the classic principal-agent model solution given in Proposition 1.2 applies. As the production technology improves, the unit price decreases and the client’s profit increases. This increase in the client’s profit makes it more attractive for the client to intervene if the client observes shirking by the vendor. After a particular level of improvement (i.e., the tipping point beyond which condition T1 is satisfied), the client pays more to the vendor, making intervention less attractive. Accordingly, the unit price decreases and the vendor’s extra expected profit over and above the reservation profit also decreases up to the tipping point. Beyond the tipping point the unit price and the vendor’s extra expected profit over and above the reservation profit increases with improvements in production technology. It follows that the client’s expected profit increases with improvements in production technology but only up to the tipping point; beyond the tipping point the client’s profit decreases with improvements in production technology. Overall, improvements in production technology have a nonmonotonic impact on unit price, the vendor’s expected profit, and the client’s expected profit. Outsourcing the design action The solution to the program when the design action is outsourced depends on the relative cost of the design and production actions and is given by:

20.

This additional payment by the client can also be interpreted as a “bribe”. Irrespective of the label for the additional payment, the bottom line is that the client makes it less attractive for him to intervene.

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T2 ¼ DV  ½aH qLLV =ðqHLV  qLLV Þ  0: D

1073

ðT2Þ:

Note that when (T2) is not satisfied the production action is more costly than the design action, and vice versa (loosely speaking). When an activity is more costly, it is likely to be more important for cost management purposes. Accordingly, condition T2 captures the relative importance of design and production actions. Solution when production action is more costly than design action The solution to the program is characterized in the following proposition when (T2) is not satisfied. PROPOSITION 3. When the design action is performed by the vendor and the production action is more costly than the design action, the solution to the program is characterized as follows. (a) If the effectiveness of the production action is sufficiently large, then the client’s commitment problem is severe and the vendor receives an expected payoff greater than his reservation profit. Technically, if (T2) is not satisfied and (T3) is satisfied, then the unit price, sV1, is determined by the (ICC) constraint and the vendor’s expected profit, UV1, is greater than the reservation profit of zero. The solution is given below. sV1 ¼ x  ½bH =ðqLHV  qLLV Þ; WV1 ¼ ½bH qHLV =ðqLHV  qLLV Þ; UV1 ¼ fx  ½bH =ðqLHV  qLLV ÞgqHLV  aH  DV [ 0; x  xT3 ¼ ½bH =ðqLHV  qLLV Þ þ ½aH =ðqHLV  qLLV Þ

ðT3Þ:

(b) If the effectiveness of the production action is not sufficiently large, then the client’s commitment problem is not severe and the vendor receives an expected payoff greater than his reservation profit. Technically, if (T2) and (T3) are not satisfied, then the unit price, sV2, is determined by the (ICV) constraint and the vendor’s expected profit, UV2, is greater than the reservation profit of zero. The solution is given below. sV2 ¼ ½aH =ðqHLV  qLLV Þ; WV2 ¼ fx  ½aH =ðqHLV  qLLV ÞgqHLV ; UV2 ¼ ½aH qLLV =ðqHLV  qLLV Þ  DV [ 0: The intuition is similar to that discussed with Proposition 1, with the obvious difference that the probability of a good unit is conditioned on the design action being taken by the vendor instead of the client. As such, condition T3 is similar to condition T1. The client’s commitment problem is severe when the effectiveness of the production action is large and the client has to pay the vendor extra to make it unattractive to intervene. Solution when production action is less costly than design action The solution to the program is characterized in the following proposition when (T2) is satisfied. PROPOSITION 4. When the design action is performed by the vendor and the production action is less costly than the design action, the solution to the program is characterized as follows.

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Contemporary Accounting Research (a) If the effectiveness of the production action is sufficiently large, then the client’s commitment problem is severe and the vendor receives an expected payoff greater than his reservation profit. Technically, if (T2) and (T4) are satisfied, then unit price, sV3, is determined by the (ICC) constraint and the vendor’s expected profit, UV3, is greater than the reservation profit of zero. The solution is given below. sV3 ¼ x  ½bH =ðqLHV  qLLV Þ; WV3 ¼ ½bH qHLV =ðqLHV  qLLV Þ; UV3 ¼ fx  ½bH =ðqLHV  qLLV ÞgqHLV  aH  DV [ 0; x  xT4 ¼ ½bH =ðqLHV  qLLV Þ þ ½ðaH þ DV Þ=ðqHLV Þ

ðT4Þ:

(b) If the effectiveness of the production action is not sufficiently large, then the client’s commitment problem is not severe and the vendor receives an expected payoff equal to his reservation profit of zero. Technically, if (T2) is satisfied and (T4) is not satisfied, then unit price, sV4, is determined by the (PC) constraint and the vendor’s expected profit, UV4, is exactly equal to the reservation profit of zero. The solution is given below. sV4 ¼ ½ðaH þ DV Þ=ðqHLV Þ; WV4 ¼ xqHLV  aH  DV ; and UV4 ¼ 0: Here again, the intuition is similar to that discussed with Proposition 1, with the obvious difference that the probability of a good unit is conditioned on the design action being outsourced and the moral hazard problem with respect to the production action being completely mitigated (i.e., condition T2 is satisfied). As such, condition T4 is similar to condition T1 with the vendor’s cost-to-productivity ratio being defined to represent the condition that the moral hazard problem is completely mitigated when condition T2 is satisfied. When the client’s commitment problem is not severe, the client will be able to use the contractible DV to mitigate the vendor’s moral hazard problem. The solution in Proposition 4.2 shows that the client can effectively mitigate the vendor’s moral hazard problem and extract all the rent. The vendor’s expected unit price covers the cost of design and production actions. By outsourcing this extra task, the client can get the production action “for free” in terms of agency costs, that is, extra payments to induce the high production action. This is a simplified version of the effect that Saouma (2008) demonstrates. Effect of improvements in production technology, qHLV and qLHV In this regime, similar to the no-outsourcing regime, we represent improvements in production technology as follows. We start from q ¼ qHLV [ qLHV ¼ q and denote the quality improvements by e  0. As before, we assume that when ε = 0 the client’s commitment problem is not severe, that is, Propositions 3.2 and 4.2 apply. The following proposition characterizes the effect of improvements in production technology. PROPOSITION 5. When the design action is performed by the vendor, q þ e ¼ qHLV [ qLHV ¼ q þ e with ε  0 and at ε = 0 the vendor’s moral hazard problem is more severe than the client’s commitment problem, that is, conditions T3 and T4 are not satisfied; as e increases, ceteris paribus, the solution changes in the following manner. (a) The domain over which the client’s commitment problem is severe increases, that is, ½d xT3 =de\0; ½d xT4 =de\0. (b) The unit price and vendor’s expected profit weakly decrease until condition T3 or T4 is satisfied and then increase, while the client’s expected profit increases until

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condition T3 or T4 is satisfied and then decrease, that is, [dsV2/dε] < 0, [dsV4/ dε] < 0, [dUV2/dε] < 0, [dUV4/dε] = 0, [dWV2/dε] > 0, [dWV4/dε] > 0 for small e, and [dsV1/dε] > 0, [dsV3/dε] > 0, [dUV1/dε] > 0, [dUV3/dε] > 0, [dWV1/dε] < 0, [dWV3/dε] < 0 for large ε. Proposition 5 is similar in tenor to Proposition 2 and shows that improvements in production technology have a nonmonotonic impact on unit price, the vendor’s expected profit, and the client’s expected profit due to the client’s commitment problem.21 Comparison of the regimes We characterize the domain over which outsourcing the design action is optimal for the client.22 For this purpose, we assume that the client is the Stackelberg leader, that is, the client decides whether outsourcing or no-outsourcing is beneficial and offers a contract to the vendor accordingly. First, we consider the setting where condition T2 is satisfied, that is, we compare the solutions characterized in Propositions 1 and 4. The comparison is characterized in the following proposition. PROPOSITION 6. When the design action is more costly than the production action, that is, condition T2 is satisfied, comparing the outsourcing and no-outsourcing of design action regimes yields the following results. (a) When the client’s commitment problem is more severe than the vendor’s moral hazard problem in both the outsourcing and no-outsourcing of design action regimes, outsourcing the design action is always optimal. The unit price in the inhouse design action regime is higher than that in the outsourced design action regime. Technically, over the domain where conditions T1 and T4 are satisfied, WV3 > WC1 and sC1 > sV3. (b) When the client’s commitment problem is more severe than the vendor’s moral hazard problem in the no-outsourcing regime but not in the outsourcing of design action regime, outsourcing the design action is optimal even if the outsourced design cost is greater than the in-house design cost. The unit price in the in-house design action regime is higher than that in the outsourced design action regime when the outsourced design cost is not sufficiently higher than the in-house design cost. Technically, if condition T1 is satisfied and condition T4 is not satisfied, T2  DC \ðx  xT1 ÞqHLC , and sC1 > sV4 if then WV4 > WC1 if and only if D T2 \ðx  xT1 ÞqHLC . and only if D (c) When the vendor’s moral hazard problem is more severe than the client’s commitment problem in both outsourcing and no-outsourcing of design action regimes, outsourcing the design action is optimal if and only if the outsourced design cost is lower than the in-house design cost. The unit price in the in-house design action regime is not higher than that in the outsourced design action regime. Technically, 21.

22.

We also consider whether the client and/or the vendor can employ a mixed strategy instead of the pure strategy for their intervention and production actions, respectively to mitigate the commitment problem. Corresponding to Propositions 1.1 and 3.1, when conditions T1 and T3 are satisfied, a solution exists in which the client does not play a mixed strategy, but the vendor does. Corresponding to Proposition 4.1, when condition T4 is satisfied, a solution exists where both the client and the vendor employ mixed strategies. However, in all of these cases both the vendor and the client are better off employing the pure strategy than the mixed strategy. The intuition that the pure strategy leads to higher profits for both the client and the vendor can be gleaned from the fact that mixed strategies are min-max strategies (for example see Vakhrameev 2003). The derivations are available from the authors. It is easy to verify that the following solutions under the outsourcing and no-outsourcing of design action regimes are not possible: (a) Propositions 3.1 and 1.2, and (b) Propositions 4.1 and 1.2.

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Proposition 6 covers the complete domain when the design action is more costly than the production action, that is, condition T2 is satisfied. Proposition 6.1 states that when the client’s commitment problem is severe under both outsourcing and no-outsourcing regimes, then it is always better for the client to outsource. The intuition stems from the fact that in outsourcing the design action, the client makes the high intervention action less effective: the assumption that qLHC > qLHV is the driving force behind this result. This is a reasonable representation of a supply chain and captures the notion that when the client is not involved in the design action, then it is more likely that he will not be able to help the vendor out, if the vendor chooses the low production action. In game theory terms, by outsourcing the design action, the client signals to the vendor that his intervention action is not going to help the vendor by much. Thus, the client can decrease the unit price because the commitment problem is less severe in the outsourcing regime. Proposition 6.2 compares the domain over which, in the no-outsourcing regime, the commitment problem is more severe than the moral hazard problem and in the outsourcing regime, the moral hazard problem is more severe than the commitment problem. The condition compares the costs and benefits of outsourcing and in-sourcing the design actions. The left side of the condition is the extra cost in the outsourcing regime compared to the in-sourcing regime with respect to the design cost: outsourcing the design action has a higher direct cost than in-sourcing. The right side of the condition is the savings in the payment to the vendor for the commitment problem in the no-outsourcing regime. Proposition 6.3 compares the domain over which, in both regimes, the moral hazard problem is more severe than the commitment problem. Here the condition boils down to comparing the extra direct cost of outsourcing the design action to the benefit of reducing the rents to the vendor (the Saouma 2008 effect). Similar to the Saouma effect, by outsourcing the additional task the client can completely mitigate the moral hazard problem with respect to the production action. Thus, our model contains the Saouma effect as well as the additional commitment effect. We now consider the setting where condition T2 is not satisfied, that is, we compare the solutions characterized in Propositions 1 and 3. The comparison is characterized in the following proposition. PROPOSITION 7. When the design action is less costly than the production action, that is, condition T2 is not satisfied, then outsourcing the design action is always optimal. The unit price in the in-house design action regime is higher than or equal to that in the outsourced design action regime. Technically, (a) when conditions T1 and T3 are satisfied, then WV1 > WC1 and sC1 > sV1; (b) when condition T1 is satisfied and condition T3 is not satisfied, then WV2 > WC1 and sC1  sV2; and (c) when conditions T1 and T3 are not satisfied, then WV2 > WC2 and sC2 = sV2. Proposition 7 shows that when the design action is less costly than the production action, outsourcing the design action always helps to mitigate the client’s commitment problem. When the client’s commitment problem is severe, the intuition in Proposition 6 applies here as well: By outsourcing, the client makes his intervention action less effective and thus makes intervening less attractive. When the commitment problem is not severe, the client essentially gets the delegation action “for free” because the unit price to induce the production action covers the vendor’s design action cost as well, that is, sC2 = sV2.

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4. Discussion, empirical implications, and concluding remarks The results of the analysis provide some insights into the trend of outsourcing. To relate our results to the insights, consider the principal-agent game being repeated with changing parameter values over time. While we make our point with improving the effectiveness of the production action, the same effect will occur when only the benefit from the good units increases. Consider outsourcing the production action when the effectiveness of the production action is small, where the classic principal-agent model solution that mitigates the vendor’s moral hazard problem is obtained. That is, in previous years the benefit from a good unit was not sufficiently large compared to the effectiveness of the production action, such that the client did not find it beneficial to intervene if he observed the vendor shirking. The vendor improved the effectiveness of production technology over time, which in turn helped to improve the client’s profits. When the improvements in the effectiveness of the production action reached a tipping point, the vendor realized that he can shirk and let the client intervene. In essence, the client’s problem of committing to not intervening has become more severe in recent years. To mitigate this commitment problem, the client has to increase the unit price to ensure that the vendor does not rush to him to help out with the intervention action. By increasing the unit price, the client reduces his own profit and makes it unattractive for him to intervene. However, the client can also address the commitment problem by outsourcing the design action; outsourcing the design action diminishes the effectiveness of the client’s intervention action. As such, the client will not find it as beneficial to intervene and help out the vendor; although the extra payment he had to make to the vendor decreases, this reduction is not commensurate with the increased cost of the additional task outsourced. Overall, starting from a setting where only the production action is outsourced, and with improvements in the effectiveness of the production action technology, we come to a setting where outsourcing more tasks is optimal even though the unit price may not increase enough to cover the direct cost of the increased responsibility. We thus provide an agency theory rationale for this casually observed phenomenon in outsourcing based on the client’s commitment problem. In summary, our insight applies to settings characterized by commitment issues of nonintervention where outsourcing more tasks helps to resolve the commitment problem in the sense that outsourcing additional tasks lowers the effectiveness of intervention for earlier tasks. The electronics and information technology sectors are characterized by settings where the client can step in and help the vendor’s production. Even in our simple model, the client’s commitment problem can become more severe than the moral hazard problem for the following reasons: (a) the client’s benefit from a good unit increasing over time, (b) the direct cost of the production and intervention actions decreasing over time, and (c) the effectiveness of the production and intervention actions increasing over time. In settings where the commitment problems become more severe, our results show the following: (a) production improvements and quality improvements will have preceded outsourcing of additional tasks, and (b) the increase in unit price is not likely to cover the increase in costs of the additional tasks. Thus, broadly speaking, the increased trend in outsourcing may not only be due to vendor’s improved abilities and client’s increased bargaining power, but may also be due to the changing nature of agency issues. An empirical prediction based on the results is that companies in industries such as electronics and information technology, where changes in the parameters over time (say for example quality improvements) lead to the client’s commitment problem becoming more severe than the moral hazard problem, will have more tasks outsourced to vendors and the vendor’s profits will decrease over time. Stated differently, empirical research that fails to

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document vendor profit increases with additional tasks should examine whether any of the parameters that capture the increasing severity of the commitment problem explain the failure to document vendor profit increases. Furthermore, empirical research can examine cases/ subsamples where the bargaining power of the buyer/client has increased over time, such as the retail sector with large retailers, and contrast the outsourcing trends with sectors where the bargaining power of buyers/clients has not increased over time. Specifically, while the increased bargaining power of large retailers could explain the shrinking margins of vendors in the retail sector, for IT vendors the severity of the client’s commitment likely explains the shrinking margins of the vendors, which can be tested using the improvements in quality and the outsourcing of additional tasks. While we examine how commitment problems might result in outsourcing more tasks to vendors, future research can consider multiple vendors and examine whether creating a hierarchy of vendors can mitigate some of the commitment problems. For example, extending the insights gleaned from our analysis, the vendors most subject to the client’s commitment problem can be distanced with another layer of vendor. Thus, a hierarchy of vendors can help mitigate such commitment problems. Future research can also examine whether commitment issues can provide a rationale for increased vertical integration over time. Appendix. We outline the track of the proofs. Proof of Proposition 1 From constraints (PC), (ICV), and (ICC), the unit price s has to satisfy the following: s  [aH/ qHLC]; s  [aH/{qHLC – qLLC}]; s  x – [bH/{qLHC – qLLC}], respectively. Use qLLC > 0, to get [aH/{qHLC – qLLC}] > [aH/qHLC]: thus, (PC) is never binding. If (T1) is satisfied, then x – [bH/ {qLHC – qLLC}]  [aH/{qHLC – qLLC}], and vice versa. Thus, if (T1) is satisfied (ICC) is binding and the unit price is given by sC1 = x – [bH/{qLHC – qLLC}], and if (T1) is not satisfied (ICV) is binding and the unit price is given by sC2 = [aH/{qHLC – qLLC}]. Use {sC1, sC2} in (1) and (2) to get {WC1, UC1, WC2, UC2}. The inequality for {UC1, UC2} follows because (PC) is not binding.

Proof of Proposition 2 Use qLHC ¼ qHLC ¼ q þ e in f xT1 ; sC1 ; sC2 ; WC1 ; UC1 ; WC2 ; UC2 g and differentiate with respect to ε to establish the proposition.

Proof of Proposition 3 From constraints (PC), (ICV) and (ICC), the unit price s has to satisfy the following: s  [{aH + DV}/ qHLV]; s  [aH/{qHLV – qLLV}]; s  x – [bH/{qLHV – qLLV}], respectively. Using (T2) not being satisfied, we get [aH/{qHLV – qLLV}] > [{aH + DV}/qHLV]; thus, (PC) is never binding if (T2) is not satisfied. If (T3) is satisfied, then x  [bH/{qLHV – qLLV}]  [aH/{qHLV – qLLV}], and vice versa. Thus, if (T3) is satisfied (ICC) is binding and the unit price is given by sV1 = x – [bH/{qLHV – qLLV}], and if (T3) is not satisfied (ICV) is binding and the unit price is given by sV2 = [aH/ {qHLV – qLLV}]. Use {sV1, sV2} in (1) and (2) to get {WV1, UV1, WV2, UV2}. The inequality for {UV1, UV2} follows because (PC) is not binding.

Proof of Proposition 4 From constraints (PC), (ICV) and (ICC), the unit price s has to satisfy the following: s  [{aH + DV}/ qHLV]; s  [aH/{qHLV – qLLV}]; s  x – [bH/{qLHV – qLLV}], respectively. Using (T2) being satisfied, we get [aH/{qHLV – qLLV}]  [{aH + DV}/qHLV]; thus, (ICV) is never binding if (T2) is

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satisfied. If (T4) is satisfied, then x  [bH/{qLHV – qLLV}]  [{aH + DV}/{qHLV}], and vice versa. Thus, if (T4) is satisfied (ICC) is binding and the unit price is given by sV3 = x – [bH/{qLHV – qLLV}], and if (T4) is not satisfied (PC) is binding and the unit price is given by sV4 = [{aH + DV}/{qHLV}]. Use {sV3, sV4} in (1) and (2) to get {WV3, UV3, WV4, UV4}. The inequality for UV3 follows because (PC) is not binding; and UV4 = 0 because (PC) is binding.

Proof of Proposition 5 Use q þ e ¼ qHLV [ qLHV ¼ q þ e in f xT3 ; xT4 ; sVi ; UVi ; WVi ; g for i = 1, 2, 3, 4 and differentiate with respect to ε to establish the proposition.

Proof of Proposition 6 The following technical lemmas will be useful to see that the complete domain is characterized in Propositions 6 and 7.

LEMMA 1. If (T3) is satisfied then (T1) is also satisfied. Use qHLC = qLHC = qHLV > qLHV and qLLC = qLLV in (T1) and (T3) to establish the lemma.

LEMMA 2. If (T2) and (T4) are satisfied then (T1) is also satisfied. Use (T2) and (T4) to establish that (T3) is satisfied. Then from Lemma 1, Lemma 2 follows. Use assumption qHLC = qLHC = qHLV > qLHV to establish Proposition 6.1. T2 from (T2) Use assumptions qHLC = qLHC = qHLV and qLLC = qLLV and the expression for D and xT1 from (T1) in (WV4 – WC1) and (sC1 – sV4) to establish Proposition 6.2. Use assumptions qHLC = qLHC = qHLV and qLLC = qLLV in (WV4 – WC2) and (sC2 – sV4) to establish Proposition 6.3.

Proof of Proposition 7 Use assumptions qHLC = qLHC = qHLV > qLHV and qLLC = qLLV to establish the proposition.

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