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Essays on supply chain management : risk management and productivity spillovers Serpa, Juan Camilo 2015

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Essays on Supply Chain Management:Risk Management and ProductivitySpilloversbyJuan Camilo SerpaB.Sc., Trent University, 2009M.A., The University of British Columbia, 2010A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Business Administration)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2015© Juan Camilo Serpa 2015AbstractThis thesis comprises three independent essays on supply chain management.In the first essay we collect data on 27,000 vertical relationships to study theimportance of different channels of productivity spillovers between upstreamand downstream firms. We explore the relative influence of two types ofchannels: endogenous and exogenous. The endogenous channel measureshow a firm’s productivity is affected by knowledge transfers (arising fromcollaboration and peer-mentoring). The exogenous channels measure theextent to which productivity is influenced by the partners’ characteristics(e.g. geographic location, inventory turnover, financial leverage, etc.). Wefind that the endogenous channel is the primary source of spillovers. We alsofind that a firm’s productivity is influenced more by the operational, thanby the financial characteristics of its partners.The second essay unveils a previously unexplored role of business insur-ance in managing supply chain risk. We show that firms may strategicallybuy insurance purely as a commitment mechanism to prevent excessive free-riding by other firms. Specifically, we show that contractual incentives aloneleave wealth-constrained firms with low incentives to prevent operational ac-cidents, and firms with sufficient wealth with excessive incentives. Insuranceallows the latter firms to credibly commit to lower effort, thereby mitigatingthe incentives of the wealth-constrained firms to free-ride.The third essay explores the interplay between public policy and riskmanagement, when governments must strike a balance between safety andindustry welfare. We focus on industries where operational accidents can bedestructive and, as a result, where the cost of third-party liability is signifi-cant. Firms in these industries may be discouraged from entering the marketas a result of these costs. If entry is inefficiently low, a social planner canincentivize firms through ex ante subsidies, which defray the costs associatedwith making operations safer, or ex post subsidies, which mitigate the finan-cial damages caused by the accident. We demonstrate that when the socialplanner values reliability over market competition, it is optimal to offer exante subsidies alone. Conversely, when competition outweighs the benefitsof reliability, a combination of ex ante and ex post subsidies is optimal.iiPrefaceModified versions of Chapter 2, 3 and 4 have been submitted for publication.All essays are co-authored with Professor Harish Krishnan.Professor Krishnan was involved in the early stage of the problem for-mulation. He also provided supervision and feedback in the design of themethodology, corrected technical and editorial mistakes, and suggested im-provements in the exposition of the results. I was responsible for developingand writing most of the work found in this thesis and I take full responsibilityfor editorial and technical mistakes, if any are found.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . viii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 The Impact of Supply Chains on Firm Level Productivity 42.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Research Background . . . . . . . . . . . . . . . . . . . . . . 82.3 Measuring Productivity . . . . . . . . . . . . . . . . . . . . . 102.4 Productivity Spillovers . . . . . . . . . . . . . . . . . . . . . 132.5 Extensions and Robustness Checks . . . . . . . . . . . . . . . 282.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 The Strategic Role of Business Insurance in Managing Sup-ply Chain Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . 423.3 Model Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 433.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.5 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72ivTable of Contents4 Policy Incentives to Mitigate the Impact of Operational TortLiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . 784.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.5 Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . 904.6 Extensions, Discussion and Conclusion . . . . . . . . . . . . 935 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99AppendicesA Technical Results and Variable Definitions of Chapter 2 . 107A.1 The Olley-Pakes Approach . . . . . . . . . . . . . . . . . . . 107A.2 The Formation of Supply Chain Networks . . . . . . . . . . . 108A.3 Variables Definition and Construction . . . . . . . . . . . . . 114B Proofs for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . 116C Proofs for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . 122vList of Tables2.1 Estimates of the production function . . . . . . . . . . . . . . 122.2 Decile distribution of log-TFP . . . . . . . . . . . . . . . . . . 132.3 Summary statistics about the customer-base of each firm. . . 222.4 Summary statistics about the firm’s networks. . . . . . . . . 222.5 Table of summary statistics . . . . . . . . . . . . . . . . . . . 232.6 Model estimates for equation 2.3. . . . . . . . . . . . . . . . . . 252.7 Summary statistics about the supplier-base of each firm. . . 302.8 Model estimates for equation (2.9) . . . . . . . . . . . . . . . . . 313.1 Optimal tiered and simple contracts - Case OC . . . . . . . . 683.2 Optimal tiered and simple contracts - Case SC . . . . . . . . 683.3 Simulation of optimal tiered contracts for different damagevariances (case SC) . . . . . . . . . . . . . . . . . . . . . . . . 693.4 Simulation of optimal tiered contracts for different damagevariances (case OC) . . . . . . . . . . . . . . . . . . . . . . . 693.5 Simulation of optimal insurance contracts with deductibles. . 70A.1 Model estimates for the selection equation. . . . . . . . . . . 112A.2 Model estimates for the linear-in-means equation. . . . . . . 113viList of Figures2.1 Determinants of productivity. . . . . . . . . . . . . . . . . . . 52.2 Direct and indirect spillover effects. . . . . . . . . . . . . . . . 62.3 Illustrative example of the model parameters . . . . . . . . . 142.4 Estimation of indirect and direct spillover effects. . . . . . . . 162.5 Networks and linear independence. . . . . . . . . . . . . . . . 182.6 An illustration of Matrix H (from Figure 2.3). . . . . . . . . . 212.7 Estimates of exogenous spillovers. . . . . . . . . . . . . . . . . 263.1 Model timeline . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 First best level of insurance. . . . . . . . . . . . . . . . . . . . 523.3 Optimal parameters - case SC, for β = 0.4, x = 100,pi = 45. . 573.4 Welfare parameters - case SC . . . . . . . . . . . . . . . . . . 583.5 Optimal contract parameters - case OC, for β = 0.4, x =100,pi = 45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.6 Welfare parameters - case OC . . . . . . . . . . . . . . . . . . 603.7 Contracting regions for the general case. . . . . . . . . . . . . 623.8 Optimal contract parameters for varying levels of α. . . . . . 714.1 Model timeline . . . . . . . . . . . . . . . . . . . . . . . . . . 804.2 An illustrative example using an inverse demand function,∆(Q) = 1−Q. . . . . . . . . . . . . . . . . . . . . . . . . . . 834.3 Illustration of the model regions. . . . . . . . . . . . . . . . . 854.4 A visualization of Lemma 12 . . . . . . . . . . . . . . . . . . 884.5 Illustrative example with inverse demand function P (Q) =1−Q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.6 Optimal policy as a function of λ. . . . . . . . . . . . . . . 93A.1 A sample network, where c12 = c32 = 1 and cij = 0 every-where else. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110viiAcknowledgementsMy time at UBC has been rewarding and special. I am deeply gratefulto Professor Harish Krishnan for his continual support. His tremendousdedication during my time in the Ph.D program was determinant. Harishintroduced me to the field of supply chain management, and devoted a greatdeal of his time in helping me become a better researcher. Not only my thesis,but also my academic aptitude and writing style has improved substantiallyas a result of his effort.I would also like to thank Professor Tim Huh, Professor Mahesh Nagara-jan and Professor Ralph Winter, who served as committee members. Theyprovided invaluable suggestions for the improvement of this thesis. Theirguidance extends far beyond this document. Each of them helped me de-velop various academic skills. I am truly indebted to them.I also wish to thank Professor Thomas Lemieux and Professor RobinLindsey, who served as university examiners, and Professor Vishal Gaur,who served as the external examiner for this thesis. Their comments werevery helpful during the final stages of this thesis. Their suggestions willhelp me improve these essays during the publication process. I am veryappreciative for their kind help.Many thanks to other faculty members at the Sauder School of Businessfor their support during various stages of this process. Finally, I would liketo thank to those individuals who indirectly helped me pave the road thatled towards the end of this process. This includes the Trent University Inter-national Program, for giving me the financial resources to come to Canada,and the University of British Columbia for its continual financial supportduring graduate school. Not to mention my friends, who kept nourishing myspirit.My deepest thanks go to my family and to Hana. You were always therewhen I needed, and for this I couldn’t be more thankful. I dedicate thisthesis to you.viiiChapter 1IntroductionThis thesis explores, through three independent essays, issues related to sup-ply chain management. The first essay focuses on firm-level productivity(Chapter 2), while the remaining two essays focus on risk management is-sues (Chapters 3 and 4).Firm-Level ProductivityThe literature has shown large and persistent differences in firm-level pro-ductivity. Simply put, the economy is divided into two groups of firms: (i)those that have successfully managed to implement an efficient productionprocess and; (ii) those that are lagging tremendously in the productivitydistribution.This gap has non-trivial implications. Firm-level productivity is one ofthe most reliable indicators, not only of firm success, but also of firm survival.As Krugman (1997) puts it, “productivity isn’t everything, but in the long-run it is almost everything.” This implies that productive firms are destinedto be at the forefront of the economy. And those firms that are lagging areultimately poised to go out of business.What causes some firms to be so productive and, at the same time, otherfirms to be so unproductive? In other words, what drives firm-level produc-tivity? Researchers have shown dozens of factors that can potentially affectthis metric. For example, the level of market competition (Syverson, 2011),the level of IT (Brynjolfsson and Hitt, 2003) - even the weather conditions(Cachon et al., 2013)- can play a determinant role. The influence of supplychains partners, however, has not been carved out.In Chapter 2 we explore the impact of vertical relationships on firm-levelproductivity. The main interest of this essay, however, is not to explore sim-ply whether supply chain partners affect each other’s productivity. After all,when two firms interact in a supply chain, they engage in an intimate rela-tionship characterized by collaboration, mutual dependency and mentoring.It is thus intuitive to expect that the productivity of a firm is affected by itschoice of partners.1Risk ManagementOur main interest in this essay is to explore the issue of “how?”. Specif-ically, what are the characteristics, actions or mechanisms through whichpartners in a supply chain end up affecting each other? Or more formally,through which channels does productivity spill over along the supply chain?For example, are firms affected via the operational, geographic or finan-cial characteristics of the partners? Or, after holding these characteristicsconstant, are firms affected by virtue of interacting with more productivepartners (e.g. via knowledge transfers)?We explore this question through econometric methods. But we mustfirst overcome several econometric challenges. For example, highly produc-tive firms also tend to be operationally efficient, financially healthy, and belocated in desirable geographies. How can we then disentangle the marginalimpact of each characteristic from the others? Second, firms are likely tochoose partners based on their strategic desire to become more productive.This will give rise to endogeneity in the formation of supply chain networks,and lead to biases in our results. Our essay develops a methodology toovercome the above econometric challenges and obtain consistent results.Risk ManagementThe second part of this thesis focuses on supply chain risk management. Weplace special emphasis on high-impact/low-probability events (e.g. productrecalls, oil spills, etc.). These types of events expose firms to financial dis-tress, illiquidity or even bankruptcy. A firm, in the face of this possibility,must balance its exposure not only to operational risk, but also to financialrisk.To hedge financial risk, firms have access to financial instruments offeredby third-party institutions. For example, depending on the situation firmscan purchase business insurance or receive public subsidies. These financialinstruments allow a firm to decrease its exposure to financial risk and, atthe same time, to optimally manage its operational risk. Through two sepa-rate essays we study optimal risk management strategies in the presence ofexternal financing instruments.In Chapter 3 we explore risk management in the presence of businessinsurance. We show that insurance sometimes may serve a purely strategicrole within the supply chain. Specifically, we show that when a large (andwealthy) firm collaborates with a small (and wealth constrained) partner,contractual tools alone leave the wealth-constrained firm with inefficientlylow incentives to exert effort because it is unable to take on the appropriate2Risk Managementlevel of liability. The opposite happens with the wealthy firm. Insurancecan serve as a credible commitment mechanism that allows the wealthy firmto commit not to exert “too much” effort to prevent operational accidents.Interestingly, this mechanism improves the efficiency of the supply chain.By committing not to exert too much effort, the wealthy firm takes awaythe incentives of the wealth-constrained firm to free-ride on the “excessive”efforts of the wealthy firm. This balances the provision of effort in the supplychain and improves overall efficiency.In Chapter 4 we focus on industries characterized by the potential forinjury or harm to third parties due to operational accidents. In this typeof setting the costs of liability can be significant. Not rarely, this conditionleads to excessive firm exit in some industries, which leaves the marketsmonopolized (and decreases market inefficiency).To prevent market failures governments often resort to offering incentivesto the firms. There are two popular incentive schemes: ex ante and ex postsubsidies. Ex ante subsidies offer incentives to the firms prior to commenc-ing operations, to defray the costs of making operations more reliable. Forexample, governments can offer grants to purchase safety equipment or tohire skilled personnel.Ex post subsidies are given after firms commence operations, and areconditional on the occurrence of an accident. These types of subsidies de-crease the firm’s exposure to the ex post costs of an accident (e.g. clean upcosts, litigation costs). For example, governments sometimes offer liabilitycaps to limit the cost of the accident for the firm.In this essay we explore the conditions under which it is socially optimalfor a social planner to offer ex ante or ex post subsidies. We find that whenthe planner values reliability over competition, the socially optimal policyis to offer ex ante subsidies alone. In the converse scenario, we show, itis optimal for the planner to offer a combination of ex ante and ex postincentives.3Chapter 2The Impact of Supply Chainson Firm Level Productivity2.1 IntroductionSyverson (2011) reviews an extensive literature that shows large and per-sistent differences in firm-level productivity, even “within narrowly definedindustries.” But what explains these differences? Syverson notes two broadcategories of factors that drive firm-level productivity: (i) factors that areinternal to the firm (such as research and development, managerial practicesand talent) and; (ii) factors that are external to it (such as regulations andproduct market competition).Though researchers have made great strides in determining the drivers ofthis productivity dispersion, the role of supply chains remains unclear. Evenso, we can reasonably conjecture that supply chain relationships play a non-trivial role. After all, firms in a supply chain collaborate, communicate andinfluence each other’s processes. Consider Wal-Mart that, by implementinga sophisticated distribution system, improved the flow of information and,consequently, the efficiency of its supply chain partners (Brynjolfsson, 2003).Similarly, Dell created an integrated network that improved the ability of itssuppliers to better match supply and demand (Fillard et al., 2011).In this essay we explore the role of supply chain linkages in influenc-ing firm-level productivity, by considering two key channels through whichproductivity can spill over across firms.First, a firm can benefit from interacting with partners that have “fa-vorable” characteristics (independent of the partners’ productivity). In theliterature, these are known as exogenous channels. As a case in point, con-sider a firm’s geographic location. When a supplier is located in a favorableregion, it may be able to ship inputs more efficiently due to the existence ofbetter transportation infrastructure, commercial regulations or climatic con-ditions. A supplier’s location by itself, therefore, can affect the productivityof its customers.42.1. IntroductionFigure 2.1: Determinants of productivity.Second, firms can benefit from interacting with productive partners (in-dependent of the partners’ characteristics). This is known as the endogenouschannel. As in the Wal-Mart and Dell anecdotes, productive firms can influ-ence the operations of their peers through mentoring or collaboration. Andeven in the absence of mentoring or collaboration, firms may learn from, andadopt, the good practices of their partners.In this context, a firm’s productivity is thus affected by three types ofeffects: (i) by the firm’s own characteristics; (ii) by the characteristics of itspartners (through the exogenous channels) and; (iii) by the productivity ofits partners (through the endogenous channel) - see Figure 2.1.By estimating these three types of effects, we obtain a nuanced pictureabout how partners influence each other’s productivity. Consider, for ex-ample, a given characteristic of a firm. We can determine the direct andthe indirect impact of this characteristic on the productivity of the supplychain partners. To illustrate this point, suppose that a change in inven-tory turnover affects a firm’s own productivity. A change in the inventoryturnover can, in addition, affect the productivity of the partners in two ways:(i) directly by the change in inventory turnover (the exogenous channel) and;(ii) indirectly, because of the change in the productivity of the firm (the en-dogenous channel) - see Figure 2.2.To estimate the impact of supply chains on firm-level productivity, wecollect a sample of approximately 27,000 supply chain relationships. Thesedata are from publicly-traded firms in the U.S. and are available due to arequirement that if a customer exceeds 10% of a firm’s annual revenue, the52.1. IntroductionFigure 2.2: Direct and indirect spillover effects.firm must disclose these sales and the name of the customer. We merge thisdataset to key idiosyncratic information about each firm, from the Compu-stat database.Our estimates find that the endogenous spillover effect is significantlylarger than any of the exogenous spillover effects. This means that inter-acting with productive firms is relatively more important than interactingwith firms with “favorable” characteristics. We find several other interestingresults (some of which are highlighted below):• The size of a firm has two counteracting effects. First, larger firmsare more productive themselves (this can be explained by the presenceof scale economies). Therefore, firms indirectly benefit from interact-ing with large partners. However, holding the partners’ productivityfixed, firms directly benefit from having smaller partners. This can beexplained by the idea that a firm has more influence over the manage-rial decisions of smaller customers (or suppliers), and this can benefitthe firm.• The effect of firm age on its partners is inversely U-shaped. Previousfindings in the literature had already found an inverse U-shaped re-lationship between firm age and productivity (Van Biesebroeck, 2005;Fernandes, 2008). But we show that, in addition, the spillover effect62.1. Introductionfollows this same pattern. This implies that it is more efficient to inter-act with partners that are neither too young nor too old. Specifically,firms benefit most from interacting with partners aged 17-20 years.• Firms located in the U.S. are more productive than foreign firms, andthis indirectly benefits the partners of U.S. firms. However, keepingthe partners’ productivity levels fixed, it is beneficial to interact withforeign firms. This last result is related to the trade literature, whichsuggests that U.S. firms benefit from technology spillovers via imports(Keller and Yeaple, 2009).• While a firm’s inventory turnover affects its partners directly and indi-rectly, financial leverage only affects them indirectly. This result hintsthat a firm’s productivity is more susceptible to the operational, thanto the financial characteristics of its partners.Before arriving at the above results, we had to deal with several identifi-cation problems that afflict the “peer-effects” literature. The most prominentis the reflection problem, which arises due to two identification issues: (i) the“correlated environment problem” and; (ii) the “entanglement problem”. Thecorrelated environment problem arises because firms that form links oftenshare common geographic, economic or technological environments. Cor-related productivity levels might thus represent the impact of (unobserved)common shocks, and not the impact of spillover effects. But even when theseshocks are absent, the reflection problem does not disappear. This is dueto the entanglement problem. This problem arises because the endogenousand the exogenous channels are inherently entangled. In other words, when-ever the characteristics of a firm change so does its productivity, i.e. theseeffects are perfectly collinear. This condition makes it extremely difficult todisentangle the true impact of a firm’s exogenous characteristics from theimpact of the firm’s productivity level on the partners. Therefore, this typeof identification is often considered one the most challenging econometrictasks in the literature (Angrist and Pischke, 2009; Jackson, 2010).We use a novel identification strategy to overcome the reflection prob-lem. This strategy is based on Bramoullé, Djebbari and Fortin (2009) ,who show that we can overcome the reflection problem when the networkof relationships satisfies a “partially-overlapping” structure. Using this re-sult, we exploit the structure of the supply chain networks to identify ourmodel. Our approach uses a series of generalized two-stage least squaresestimators, which extract information from multiple echelons in the supplychain networks. Our identification approach is strong because we rely on the72.2. Research Backgroundinternal structure of our data (and not on external instruments) to deriveour estimates.To test the robustness of our main results, we re-estimate them by varyingsome model constructs. We were particularly concerned with the fact thata firm may be inclined to choose partners with specific characteristics, e.g.geographic location or firm size. Hence, we need to consider the possibilitythat our results are correlated with the choice of supply chain partners, andthis would bias our estimates. To address this concern, we re-estimate ourresults by controlling for this factor. We also vary other constructs, suchas the specification of the production function, the variable definitions, theindustry classification, etc. Our conclusions are robust to these checks.This manuscript has two main contributions. From a practical stand-point, our results help us understand how productivity is channeled acrossthe supply chain. We provide evidence about the influence of various char-acteristics, and we also show that interacting with productive partners (theendogenous channel) is the main source of productivity spillovers - more sothan any exogenous channel. Our results can help practitioners at the timeof building supply chain relationships.The second contribution is that we are (to our knowledge) the first tojointly identify endogenous and exogenous spillover effects using data on firmnetworks. To accomplish this, we overcome several identification issues byusing recent techniques drawn from the peer effects literature. We providea novel framework to estimate spillover effects across supply chains. Thisframework can guide future research on similar problems. As such, thismanuscript bridges the literature on peer effects and supply chains.The remainder of this article is organized as follows. In the next sectionwe review the literature. We then estimate the production function in §2.3and, using this function, the productivity of each firm. In §2.4 we estimatethe influence of supply chain linkages on firm-level productivity. In §2.5 westudy a few extensions and robustness checks. We conclude in § Research BackgroundThis essay is related to two streams of empirical research: (i) firm-levelproductivity and; (ii) operational performance and supply chain linkages.A survey of the literature on firm-level productivity is provided by Syver-son (2011). We can divide this literature into two groups. The first groupuses data from a single industry and, in this case, productivity is often mea-sured through inputs and outputs that are specific to the industry. For82.2. Research Backgroundexample, Kellogg (2011) collects data on the oil-drilling industry to estimateproductivity gains among firms that collaborate through long-term contracts.Cachon et al. (2013) use data from automobile plants to estimate the impactof severe weather on productivity. Chandra et al. (2012) use Medicare datato explain productivity differences across U.S. hospitals.The second group does not focus on studying the productivity of a par-ticular industry but, rather, on making conclusions that apply across largesectors of the economy. The problem is that inputs and outputs vary tremen-dously across industries. To avoid inconsistencies, researchers often gaugeproductivity using the “value-added” approach. In this approach inputs con-sist of capital and employment, and output is measured as value-added.Our essay uses this approach. To construct some variables, we follow themethodology adopted by Brynjolfsson and Hitt (2003), who use the Compu-stat dataset to measure the relationship between firm-level productivity andIT.When we measure productivity, we must worry about simultaneity andselection biases (these biases are explained in §2.3). We control for theseissues by adopting a widely used identification strategy developed by Olleyand Pakes (1996).There is also a stream of empirical research dedicated to studying theimpact of supply chain linkages on a firm’s operational performance. For ex-ample, Kalwani and Narayandas (1995) report that suppliers observe higherreturns on investment when they are engaged in long-term relationships withtheir customers. Hendricks and Singhal (2005) study the effect of supplychain disruptions on long-run stock price performance. In the global sourcingliterature, Jain, Girotra, and Netessine (2013a) study the impact of sourc-ing relationships on inventory. The authors show that when firms sourcefrom foreign suppliers, their inventory investments increase. Lieberman andDemeester (1999) study productivity growth when suppliers and manufac-turers collaborate through Just-In-Time delivery. The authors use data onthe Japanese auto industry.We contribute to this literature by studying the different channelsthrough which productivity spills over across firms. To our knowledge, thistype of analysis has not been done previously, perhaps because of the seriousidentification challenges encountered in this literature. The most promi-nent identification issue is the reflection problem, a problem first studiedby Manski (1993). The reflection problem has become a central issue whenestimating spillover effects. Fortunately, the literature has come a long wayover the past years. A review of these developments can be found in Blumeet al. (2010).92.3. Measuring ProductivityIn this essay we exploit recent results found by Bramoullé, Djebbari,and Fortin (2009), who show that it is possible to overcome the reflectionproblem if the network of relationships satisfies a particular structure (whichinvolves partially overlapping interactions). Our data satisfies the necessarystructure and, therefore, we can control the reflection problem. We use theestimators proposed by Lee (2007) to obtain our estimates. These estima-tors are asymptotically optimal for estimating spillover effects in networkinteractions.2.3 Measuring Productivity2.3.1 Econometric SpecificationWe use Total Factor Productivity (TFP) as our measure of productivity.TFP is desirable because it is invariant to the intensity of use of observableinput factors (Syverson, 2011). To obtain our estimates, we use an approachsimilar to Imrohoroglu and Tuzel (Imrohoroglu and Tuzel, 2014).Consider a log-linear Cobb-Douglas production function1yirt = αrt + βkkirt + βllirt + ρirt + εirt (2.1)where yirt is the natural logarithm of value-added for firm i in industry rand year t, while k and l represent the log of capital and labour. Parameterαrt measures the industry fixed-effect on output; we classify firms into anindustry according to their 3-digit SIC code. The term ρirt represents thefirm’s TFP, which can be interpreted as the relative productivity rank ofa firm within its industry. The term εirt is normally distributed randomshock. Therefore, if we let αˆrt, βˆl, βˆk denote the estimates of the productionfunction, we have thatρˆirt = yirt − αˆrt − βˆllirt − βˆkkirtis the estimated log TFP of firm i at time t.2.3.2 Identification StrategyTo measure log-TFP, we use the value-added approach, where inputs consistof capital and labour and output is measured as value-added. Althoughthis is one of the most popular approaches to measure productivity, two1We obtain this function after taking logarithms of Y = AKβKLβLeε.102.3. Measuring Productivitykey problems arise when estimating TFP through OLS: simultaneity andselection biases.A simultaneity bias arises because the input factors and the observedproductivity level are simultaneously determined. In these cases, there iscorrelation between the inputs and the error term. For example, if a firmobserves a favorable shock (that is unobserved by the econometrician), thefirm knows that it needs less labour to produce a given level of output. Asa result, the firm will decide to hire less, and OLS will overestimate βl.A selection bias arises because a firm’s profitability is correlated with itslevel of capital stock, which is fixed in the short term. That is, a firm withlarger capital stock is less likely to exit the market (despite low productivitydraws). This is because the firm expects to earn greater profits in the future.Given the negative correlation between capital stock and the exit probability,productivity is also correlated to the capital stock, and OLS underestimatesβk.To control for both of these biases, we use an estimation method pro-posed by Olley and Pakes (1996). We explain this methodology in AppendixA. Although this approach has become a standard one in the productivityliterature, a drawback is that we must use positive capital investments as aproxy variable. In principle the OP approach can still be used in these situ-ations, although it requires that we ignore all data points with non-positivecapital investments. Fortunately, in our sample less than 2.6% of the firmsreport non-positive capital investments.2There are other “garden variety” problems with the value-added approach(see Griliches and Mairesse 1997). For example, there is lack of informationon the quality dimensions and on the utilization of these variables, and pro-ductivity is sensitive to output prices. Fortunately, productivity estimatesare “likely quite robust to [these] measurement peculiarities. The inherentvariation in establishment micro-data is so large as to swamp any smallmeasurement-induced differences in productivity metrics (Syverson, 2011).”As such, we treat these problems as minor data limitations (like virtuallyevery paper in the literature). But in this point we do more than assuming,and we test the robustness of our productivity estimates in § some studies, the Olley-Pakes approach is undesirable because the proportion ofobservations with non-positive capital investments is large. To deal with this problem,Levinsohn and Petrin (2003) created a very similar methodology that uses intermediatematerials (instead of capital investments) to solve the simultaneity bias. This methodol-ogy, however, does not control for the selection bias.112.3. Measuring ProductivityElasticity (log) Olley-Pakes OLSCapital (βk)0.4492 0.4138(0.0041) (0.0019)Labour (βl)0.5986 0.6305(0.0067) (0.0016)Scale elasticity 1.0191 1.0435Table 2.1: Estimates of the production function2.3.3 Data DescriptionTo estimate TFP, we use data from the Compustat database, which containsfinancial information about publicly-traded firms. Each firm-year observa-tion reports data on annual sales, levels of inventory, assets, liabilities, etc.We use datasets from the NBER-CES Manufacturing Industry Database toretrieve price deflators for capital stock, materials and output. We also re-trieve data from the U.S. Bureau of Labor Statistics to obtain labour relateddata (i.e. average compensation costs and deflator prices). Our methodologyto construct the variables can be found in the Appendix section.Before building our dataset, we applied a filtering mechanism to dis-card useless observations. First, we dropped all observations reported before1962 (due to well-known reporting biases). We also discarded firms withSIC code 99, as these are unclassified establishments. Because TFP is anindustry-specific measure, these observations are useless. Finally, we deletedobservations reporting non-positive value added, number of employees, cap-ital investment gross property, sales or plant and equipment. Using the dataabove, we constructed an unbalanced panel containing 22,133 distinct firmsand 137,864 firm-year observations. Note that, because we use data fromCompustat, our analysis focuses on publicly-traded firms.2.3.4 EstimatesWe report the elasticity estimates in Table 2.1 (using the Olley-Pakes ap-proach). For comparison purposes, we also report the OLS estimates of theproduction function. Though not reported, our regression includes dummiesto control for industry fixed effects.Our results show that OLS underestimates the capital stock elasticityand overestimates the labour elasticity. These results allow us to appreciatethe impact of simultaneity and selection biases that we discussed earlier. Wecan also observe that firms face (mild) increasing returns to scale, given that122.4. Productivity SpilloversPercentile10 20 40 60 80 90log(TFP) 0.281 0.436 0.579 0.69 0.82 0.94490th − ithpercentile ratio 1.941 1.662 1.44 1.289 1.132Table 2.2: Decile distribution of log-TFPthe scale elasticity is greater than one (i.e. βk+βl> 1).As mentioned above, we test the robustness of our estimates in §2.5.3.In addition, we compared our results with those found in similar studies. Tothis end, we looked at the decile distribution of log(TFP) and the inter-decileproductivity ratio (see Table 2.2). This ratio determines the degree to whicha firm in the 90th percentile of the productivity distribution is more efficientthan an nth percentile firm. For example, a ratio equal to two would implythat a 90th percentile firm produces twice as much output (with the samedegree of measured inputs) as an nth percentile plant. Syverson (Syverson,2004) reports that the 90th − 10th percentile ratio is equal to 1.92, a ratiothat is almost identical to ours. In our case, this ratio is equal to 1.941.Imrohoroglu and Tuzel (Imrohoroglu and Tuzel, 2014) report a 90th − 20thdispersion of 1.8, while, in our case, the dispersion is equal to 1.662.2.4 Productivity SpilloversIn this section, we construct an econometric model to jointly estimate theendogenous and the exogenous spillover effects on TFP. Due to data limi-tations (which are explained in §2.4.5), we restrict our main analysis to thefirm’s customer-base. In §2.5.1, however, we estimate spillover effects for afirm’s supplier-base.The modeling presented below borrows terminology from Sacerdote(2001) and from Bramoullé, Djebbari, and Fortin (2009).2.4.1 Setup and NotationConsider a sample of N firm-year observations, N = {1, 2, . . . , N}. Each i ∈N is characterized by a value for TFPi and a K-vector Xi = [x1i, . . . , xKi] ofcharacteristics that could affect TFPi. Each observation is also characterizedby a set of customers. Let C be an N ×N matrix with typical element cij ,where cij equals one if j is a customer of i, and zero otherwise. We assumethat cii = 0. Let W be an N × N matrix characterizing the strength of132.4. Productivity SpilloversFigure 2.3: Illustrative example of the model parametersthe relationships between a firm and each of its customers. This matrix hastypical elementwij =cij ·(salesij)∑Nj=1 cij ·(salesij)if cij = 10 otherwisewhere salesij is the revenue received by firm i from customer j. Using graphtheory, we can represent firms as nodes, and their relationships throughweighted edges. As such, W is the weighted adjacency matrix of the graph.Two firms belong to the same network if they can be connected throughan undirected path of relationships, that is, if they are (weakly) connectedin the graph. We use ψ to index the networks, and assume that there areΨ networks in total, each with size Lψ.3 See Figure 2.3 for an illustrationfor a network with N = 7 firms and Ψ = 2 networks (the networks have sizeL1 = 4 and L2 = 3 respectively).3Because each i ∈ N represents a firm-year observation, we may be able to find twonetworks that are comprised by the same firms, but at different points in time.142.4. Productivity Spillovers2.4.2 Econometric SpecificationWe specify a linear-in-means model, whereTFPi = β + (γ1x1i + ...+ γKxKi)︸ ︷︷ ︸firm effects+(θ1xC1i + ...+ θKxCKi)︸ ︷︷ ︸exogenous spillovers(2.2)+ ωTFPCi︸ ︷︷ ︸endogenous spillover+ uiHere, xCki≡∑j wijxkj denotes the weighted average characteristics of i’s cus-tomers and TFPCi ≡∑j wijTFPj denotes their average productivity. Theerror term is represented by ui, which we assume to be normally distributedwith E [ui|Xi,Wi] = δXi for some K-vector δ.The vector γ = [γ1, ..., γK ]′ represents the firm effects; θ ≡ [θ1, ..., θK ]′represents the exogenous spillover effects, i.e. the impact of a customer’scharacteristic on a firm’s TFP and; ω represents the endogenous spillovereffect, i.e. the impact of a customer’s TFP on a firm’s TFP. Finally, δrepresents the correlated effects. If δ 6= 0, this implies that the customersand the firm are affected by common (economic, geographic or technological)shocks.If we let TFP=[TFP1.....TFPN ]′, X = [X1...XN]′ and 1 be an N×1vector of ones, the matrix representation of our model isTFP =1β + Xγ + WXθ + (W)TFPω + u (2.3)where E[u|X,W] = X′δ.These estimates give very detailed descriptions about the spillovers ef-fects. We can estimate the direct impact of a customer’s kth characteristicon its supplier (through the exogenous spillover effect, θk). We also estimatehow a given characteristic affects a firm’s own TFP (through γk), and inturn, how this change indirectly affects the supply chain partners throughthe endogenous effect, ω. Therefore, the indirect spillover effect is γkω. Forexample, in figure 2.4 we look at a scenario where the firm has one cus-tomer. If firm j increases in size (by one unit), i’s TFP will receive a directspillover equal to θsize. At the same time, a unit change in the size of thecustomer will affect the productivity of this customer by γsize. This change,in turn, will affect i’s TFP firm through the endogenous effect, ω. The sizeof the indirect spillover effect is γsizeω. Therefore, the total spillover effectis θsize + γsizeω.152.4. Productivity SpilloversFigure 2.4: Estimation of indirect and direct spillover effects.Unfortunately, the estimation of these effects is considered one of themost daunting econometric tasks, and in many cases an impossible one. Oneof the most challenging problems is the reflection problem.2.4.3 The Reflection ProblemIn his seminal paper, Manski (1993) shows that our linear-in-means modelcannot be identified through OLS. This is because of two identification is-sues: (i) the “correlated environment” problem and; (ii) the “entanglement”problem. The first issue arises because, in a supply chain, firms often sharecommon geographic, economic or technological environments. Correlatedshocks are present when δ 6= 0. If the presence of these effects is not con-trolled for, the econometrician may establish a spurious relationship betweenthe productivity of a firm and the firm’s customers.But even when this problem is absent, i.e. δ = 0, it is not possibleto identify our model through OLS. This is because of the entanglementproblem, which is the more complicated problem. This issue arises becausethe endogenous effect cannot be disentangled from the exogenous effects. Tosee this, observe that given equation (2.3), we can writeE[TFP|X,W] = 1β + E[X|W]γ + WXθ + E[(W)TFP|X]ω (2.4)Therefore, we can eliminate (W)TFP by rewriting (2.3) asTFP = β(I− ωW)−11 + (I− ωW)−1(Xγ + WXθ) (2.5)+(I− ωW)−1u= 1b1+Xb2 + WXb3 + uo (2.6)162.4. Productivity SpilloversEquation (2.5) is known as the reduced form equation. The reduced formequation and equation (2.3) are informationally equivalent.The reduced form estimates (b1,b2,b3) cannot be mapped onto(β, γ, θ, ω) because there is insufficient information. This is because theTFP and the exogenous regressors are linearly dependent, i.e. as the exoge-nous characteristics vary so does TFP. For this reason, the exogenous effectscannot be identified apart from the endogenous effects.Fortunately, the reflection problem does not completely close the door foridentification. There are some alternatives to dealing with this problem. Inthis essay we use a novel identification strategy that exploits partially over-lapping network interactions. This approach uses the structural informationof our model instead of relying on exogenous instruments. This feature makesour estimation approach particularly strong.2.4.4 Identification StrategyBramoullé, Djebbari, and Fortin (2009) show that identification is possibleif the network interactions satisfy a structure involving partially overlapping“peer” groups. This is because each agent has a peer group that is differentfrom the peer group of every other agent. Under some conditions (that weexplain below), we can exploit these differences by instrumenting the exoge-nous “peer of peer” effects on the peer effects, which allows us to disentanglethe exogenous effects from the endogenous effects. This approach does notimpose stringent assumptions on the structure of the data, which makes ouridentification technique robust.Bramoullé, et al.’s approach can be applied under two scenarios: onewhere correlated effects are absent, and one where the correlated effects arefixed across the network. We derive our estimates under these two assump-tions.No Correlated EffectsIn this section, we assume that δ = 0 or, equivalently, that E[u|X,W] =0.In the absence of correlated effects, Proposition 1 in Bramoullé, Djebbariand Fortin (2009) show that if the identity matrix, I, and the matrices Wand W2 are linearly independent, the reduced form estimates from equation(2.6) can be mapped onto (β, γ, θ, ω). This means that our model can beidentified. For example, in network (a) from Figure 2.5 identification isnot possible because I,W and W2 are linearly dependent. In network (b)identification is possible because the matrices exhibit linear independence172.4. Productivity SpilloversFigure 2.5: Networks and linear independence.The intuition behind this result is the following: matrixW2 describes theweighted relationships between a firm and the “customers of its customers.”4If I, W and W2 are linearly independent, then any WnX (for n = 2, 3,...)can serve as an identifying instrument. That is, the third or fourth echelonof customers can also be used to instrument for the first echelon.In Figure 2.5 we illustrate this insight. In the network from Figure 2.5(a)identification is not possible because there is a complete overlap in the net-work.5 In other words, all firms derive one third of their revenue from eachpartner. Conversely, in the network from Figure 2.5(b) it is possible to iden-tify the model. This is because the (partially overlapping) network structureof 2.5(b) allows us to extract the unique impact of a firm on its partners.For example, the “influence” (i.e. the weight) of Firm 2 is higher on Firm 4than on Firm 1; the influence of Firm 4 is higher on Firm 3 than on Firm1, etc. We exploit these differences to tease out the exact impact of eachcharacteristic on the supply chain partners.To check that I,W and W2 are linearly independent, we use Corollary 1in Bramoullé, Djebbari and Fortin (2009). According to this corollary, linearindependence is guaranteed if the diameter of the network is greater than4Similarly,W3,W4, ... represent the relationship between the firms and lower customerechelons in the network.5The “star” and “ring” networks, and bipartite networks are other types of structurescharacterized by this type of linear dependence.182.4. Productivity Spilloversthree; from the data, we found that the diameter of our network is twelve.6Hence, we can identify our estimates.After verifying linear independence, we proceed to identify equation (2.3).To this end we use a series of generalized Two-Stage Least Squares (2SLS),based on an approach proposed by Lee (Lee, 2007). This estimator is desir-able because it is asymptotically optimal under i.i.d. errors.7 To apply thisestimator, we follow two steps:Step 1 We begin by estimating an over-identified 2SLS model. Weuse X, WX, and W2X as first-stage instruments (to instrument for(W)TFP). Then, we use (W)TFP, X and WX as regressors in thesecond-stage equation. This specification allows us to obtain the estimates(β2SLS , γ2SLS , θ2SLS , ω2SLS).Step 2 We use the estimates obtained in step 1 to estimate the expectationof equation (2.5)- the reduced form equation:E[(W)TFP2SLS|X,W]= W(1− ω2SLSW)−1[β2SLS1 + Xγ2SLS + WXθ2SLS]Next, we specify the second 2SLS model. This model usesE[(W)TFP2SLS|X,W], X and WX as instruments (once again, to in-strument for (W)TFP). We use (W)TFP, X and WX as second-stage regressors. After regressing this model, we obtain the Lee estimates(βLee, γLee, θLee, ωLee).Correlated EffectsIn the presence of correlated effects (i.e. when δ 6= 0), the estimators aboveyield biased estimates. To deal with this problem we consider, for a givennetwork ψ, the following equationTFPψ = βψ1 + Xψγ + WXψθ + ω(W)TFPψ + uψ (2.7)6Note that the diameter of a network is defined as the “longest shortest path” betweenany two vertices. Therefore, to satisfy Corollary 1 in Bramoullé, Djebbari and Fortin(2009), it suffices to find a “shortest path” that is larger than three.7Our model controls for heteroskedasticity, i.e. we use the generalized version of Lee’sEstimators. As a result, our estimator loses the optimality property, but maintains itsconsistency property. The loss of optimality does not seem to make a significant difference,given that our model estimates are very robust.192.4. Productivity SpilloversHere, βψ represents unobserved shocks that are common to each memberof the network. We thus replace the assumption that E[uψ|Xψ] = 0 bythe weaker assumption that E[uψ|Xψ] = X′δ but E[uψ|Xψ,βψ] = 0. Thisidentification strategy assumes that correlated effects are present within themembers of network ψ. In other words, we allow for the possibility thatmembers of a given network are affected by common shocks (e.g. climatic,technological, economic, etc.).8To identify equation (2.7), we define the square matrixH of sizeN , whereeach entry hij = 1Lψ if i, j ∈ ψ and hij = 0 otherwise. This matrix allows usto average out the fixed network effects (see figure 2.6 for an illustration).Using this matrix, we transform our model by letting(I−H)TFP = (I−H)Xγ + (I−H)WX (2.8)θ + ω(I−H)(W)TFP + (I −H)uProposition 5 in Bramoullé, Djebbari, and Fortin (2009) shows that whenevermatrices I, W, W2and W3 are linearly independent, then equation (2.8)can be identified. This condition is more stringent than the case wherecorrelated effects are absent. This is due to the fact that, to control forcorrelated effects, some information is lost. Because the diameter of ournetwork is greater than 3, Corollary 1 in Bramoullé, Djebbari and Fortinalso guarantees linear independence of these matrices.To identify our model in the correlated effects case, we again use Lee’sestimators. In this case, the first step of our estimation consists in specifyinga 2SLS regression. We use (I−H)X, (I-H)WX and (I−H)W2X as first-stage instruments, to instrument for (I−H)(W)TFP. In the second-stageregression, we use (I−H)(W)TFP,(I−H)X and (I−H)WX as regres-sors. This model allows us to recover estimates γ2SLS, θ2SLS and ω2SLS .In the second step, we use these estimates to obtain the expectation ofthe reduced form equation, whereE[(I−H)(W)TFP2SLS|X,W]= W(1− ω2SLSW)−1[(I−H)(Xγ2SLS + WXθ2SLS)]We then perform a second 2SLS regression, but this time we use instrumentsE[(I−H)(W)TFP2SLS|X,W], (I−H)X and (I−H)WX in the first-stage equation. In the second-stage, we use (I−H)(W)TFP, (I−H)Xand (I−H)WX as regressors.8Recall that networks are formed across time periods. For this reason, βψ also includestime fixed-effects.202.4. Productivity SpilloversH=14141414 0 0 014141414 0 0 014141414 0 0 014141414 0 0 00 0 0 131313130 0 0 131313130 0 0 13131313Figure 2.6: An illustration of Matrix H (from Figure 2.3).2.4.5 Data DescriptionThe statement No. 14 of the Financial Accounting Standards Board (FASB)requires firms to disclose the revenue derived from sales to customers thatexceed 10% of their annual revenue. The reports contain information aboutthe principal firm (i.e. global identifier and company name), the year, thecustomer’s name and the sales made to this customer. Compustat retrievesthese relationships from annual 10-K filings, and stores this information inthe business segments database. Note that this dataset only reports “major”customers, i.e. customers that exceed the 10% threshold. While it wouldhave been desirable to have data on all customers, this is not a severe limita-tion. This is because very small customers are unlikely to play a significantinfluence on the firm. That is, supply chain relationships are important inso far as they represent a significant portion of a firm’s annual revenue (e.g.more than 10%).The source dataset is considerably messy and the reports do not containglobal identifiers for the customer. To build this dataset, our first step wasto perform a visual check. In this check, we noticed that some observationsreported ambiguous statements about a customer, for example, by referringto it as “customer 1”, or by reporting “2 customers” instead of detailing theirnames. In other cases, the firm’s name was ambiguously abbreviated. Ifan observation fell into any of these cases, it was discarded. In the vastmajority of the cases, however, the customer’s name was fully spelled, or theabbreviation was clear enough. For example “Johs. & Johs.” was clearlymaking reference to “Johnson & Johnson”. In these cases, the observationwas rewritten to match the original company name.After performing our visual check, we matched the firms through a pho-netic string algorithm. This algorithm allowed us to match the names re-ported by a firm to the Compustat database. If the word matching softwarewas unable to properly match the company, or if the accuracy of the match212.4. Productivity Spillovers# of customers observations (%) Avg. revenue0 8,013 (29.27%) 01 13,981 (51.07%) 19.89%2 3,380 (14.17%) 30.71%3 1,185 (4.33)% 43.06%≥4 318 (1.15%) 48.93%Table 2.3: Summary statistics about the customer-base of each firm.Network Size (Lψ)Mean 286.56Std. Deviation 281.84Median 256Minimum 2Maximum 785Number of Networks(Ψ) 4485Table 2.4: Summary statistics about the firm’s networks.was below 90%, the firm was manually matched. Our manual match wassuccessful in the vast majority of these cases.After obtaining TFP estimates, we matched each firm to each of itscustomers. If there were data missing about the relationship, we discardedthe customer. For example, in some cases the firm did not report the amountof sales made to one of its customers, and in other cases this amount wasequal to zero or negative.After cleaning the data, we kept all firm-year observations that eitherreport a customer, or those that are reported by a customer. The resultingdataset contains 27,699 firm-customer relationships, N = 26, 336 firm-yearobservations and Ψ = 4, 485 networks ranging between 1976 and 2009. Ourdataset covers 6,597 firms, which is approximately 28% of the Compustatuniverse. We use these firms to construct our interaction matrix W.Table 2.3 summarizes the customer-bases for all firms; Table 2.4 sum-marizes the network characteristics. From Table 2.3 we can observe that29.27% of our sample consists of firms without “major” customers. Most ofthese firms are retailers, e.g. Wal-Mart and K-Mart or apparel stores, orconsumer services firms. Approximately half of the firms report exactly onemajor customer. This customer represents, on average, 19.89% of the firm’sannual revenue. About 14% of the firms report two major customers, andthese customers represent 30.71% of the firm’s annual revenue. On average,222.4. Productivity Spillovers(1) (2) (3) (4) (5) (6) (7)VARIABLES N mean std.dev. p25 p50 p75 p99TFP 27,376 1.783 0.649 1.505 1.782 2.094 3.386Total assets 27,376 6,854 33,901 41.69 249.9 2,136 116,672Leverage 27,376 0.267 0.314 0.0763 0.227 0.373 1.159Inv. Turnover 27,376 0.140 0.182 0.0543 0.120 0.190 0.518Age 27,376 17.12 14.29 6 12 26 55Region: West 27,376 0.223 0.416 0 0 0 1Region: Midwest 27,376 0.213 0.409 0 0 0 1Region: South 27,376 0.247 0.431 0 0 0 1Region: Northeast 27,376 0.351 0.477 0 0 1 1Table 2.5: Table of summary statisticswe capture 25.5% of the annual revenue from firms that report at least onecustomer.9 From table 2.4 we can observe that the average customer belongsto a network of size 287, and that the largest network includes 785 firms.10Vector of Characteristics To build the vector of firm characteristics(Xi) we include firm age, size, financial leverage, inventory turnover andgeographic region (at the beginning of the period). These characteristics aresummarized in Table 2.5.We proxy for firm age by observing the year in which the firm first ap-peared in Compustat; we also include the square of firm age to control for(observed) non-linearities. To control for the effect of firm size, we use thenatural logarithm of total assets. We include financial leverage and inventoryturnover to proxy for the financial and operational conditions of the firm.Financial leverage is defined as the ratio of total debt to the book value oftotal assets, and inventory turnover is defined as the ratio of annual net-salesto average inventory.11 We also control for the square of inventory turnover9Note that a significant fraction of the sales (which are not reported in these data) goto small consumers.10To ensure that our estimates (particularly when correcting for network fixed effects)are not affected by the inclusion of small networks, we re-estimated them by excludingthese networks. As we show in §2.5.3, our results are robust to the inclusion or exclusionof small networks.11A more precise definition for inventory turnover uses the ratio of the costs of goodssold to inventory. However, some observations were missing information about the “costof goods sold”. But as we show in §2.5.3, our results are invariant from the choice ofdefinition.232.4. Productivity Spilloversafter observing a non-linear relationship between TFP and turnover. Fi-nally, we control for geographic effects by dividing firms into five regions:Northeast, Midwest, South, West and Overseas. The first four categoriesrepresent the official Census-Bureau designated regions, and the (excluded)dummy Overseas includes international firms, or those firms located outsideof mainland U.S. Approximately, 5% of the firms are located overseas.The definitions above are based on standard accounting definitions andthose used by similar studies (e.g. Patatoukas, 2011; Keller and Yeaple, 2009,Kalwani and Narayandas, 1995). However, some variables can be defined inalternative ways. To ensure that our results are not skewed by a particulardefinition, we re-estimated our model using alternative definitions. Theserobustness checks are explained in § EstimatesWe show our estimates in Table 2.6: in Column 1, we report the OLS es-timates; in Column 2, we present the Lee estimates (without controls forcorrelated effects); in Column 3, we present the Lee estimates with controlsfor correlated effects.12 A Hausman test reveals that the model in Column3 yields the most robust estimates (the p-value for this test is smaller than0.001). We summarize our main findings below.Endogenous Effect (ω): The Lee estimators report the presence oflarge and positive endogenous effects. The estimate of the coefficient, ω,is equal to 0.5979 when we do not control for correlated network effects;the coefficient is equal to 0.6081 when we correct for this source of bias.13According to this result, if the average log-productivity of the customer-baseincreases by one standard deviation, the firm’s log-TFP increases by about3/5ths of a standard deviation.Note the OLS estimators show that the endogenous effects are negative.That is, when we fail to control for the reflection problem, our results showthat the productivity of a firm decreases when it has a more productivecustomer-base. This comparison allows us to appreciate the severity of thisidentification issue.Exogenous Effects: Below we examine the impact of the exogenouscharacteristics (we focus on interpreting the most robust estimates, i.e. theestimates from Column 3). We explain both the firm effects, γ, and thespillover effects, θ. Note that we illustrate the direct (θ) and indirect (ωγ)spillover effects in Figure 2.7.12In all three models, TFP is estimated through the Olley-Pakes approach.13Both of these estimates are robust at the 1% level of confidence.242.4. Productivity SpilloversEffect Variable(1) (2) (3)OLS Lee LeeCoeff. t-stat Coeff. t-stat Coeff. t-statEndogenous effect (ω) Cust. TFP -0.0926** -2.38 0.5979*** 3.02 0.6081*** 6.35Age 0.0079*** 3.02 -0.0027 -0.72 0.0101*** 6.19Age2 -0.0002*** -5.04 0.0001* 1.96 -0.002*** -5.73Exogenous Size 0.0354*** 4.55 -0.0660** -2.46 -0.0567*** -4.42spillover Leverage -0.0000 -0.41 0.0000 0.89 0.0001 0.72effects (θ) Inv. Turnover 0.9955*** 10.00 -0.0313 -0.91 0.3551*** 3.09Inv. Turnover2 -0.2027*** -6.81 0.0169 0.61 -0.0085*** -3.00Region: West 0.3579*** 8.88 -0.3982** -2.20 -0.2621*** -3.01Region: Midwest 0.2638*** 10.09 -0.1819** -2.48 -0.0297 -0.66Region: South 0.2636*** 9.06 -0.1048 -1.12 -0.0374 -0.84Region: Northeast 0.2719*** 10.68 -0.1959** -2.56 -0.0217 -0.50Age 0.0079*** 4.08 -0.0167*** -11.27 0.0042*** 3.63Age2 -0.0003*** -9.20 0.0003*** 9.89 -0.0002*** -7.12Size 0.1346*** 58.67 0.0489*** 9.06 0.1396*** 76.84Leverage -0.0559*** -2.99 -0.1619*** -7.26 -0.0592*** -4.16Firm Inv. Turnover 0.4619*** 8.39 -0.1611*** -2.74 0.3989*** 9.68effects (γ) Inv. Turnover2 -0.0598*** -7.11 0.0336** 2.51 -0.0491*** -6.88Region: West 0.6554*** 23.07 0.1000*** 5.68 0.5838*** 33.63Region: Midwest 0.5267*** 21.60 0.0656** 2.44 0.5199*** 33.36Region: South 0.3844*** 23.09 0.0508*** 3.87 0.3594*** 28.16Region: Northeast 0.4890*** 25.34 0.0835*** 5.42 0.4523*** 33.47Intercept 1.4819*** 19.51Number of Observations 27,376 27,376 27,376R-Squared 0.8920 0.0003 0.8787Fixed Effects (Network level) Yes No Yes*** p<0.01, **p < 0.05,*p <0.1Table 2.6: Model estimates for equation Productivity SpilloversFigure 2.7: Estimates of exogenous spillovers.Size The exogenous firm effect is positive (γsize = 0.1396). Thismeans that larger firms are more productive, which is likelydue to scale economies in the production processes (see §2.3).As a result, firms indirectly benefit from interacting with largercustomers. Note that the size of the indirect spillover effectis γsizeω = (0.1396) (0.6081) = 0.085. In contrast, the coeffi-cient estimate for the direct (exogenous) spillover effect is nega-tive (θsize = −0.0567). This means that, holding the partner’sproductivity fixed, firms benefit from having smaller customers.This result can be explained by the idea that when a firm con-tracts with smaller customers, it can exert more “influence” overthem (and this can benefit the firm). The indirect effect dom-inates the direct spillover effect, implying that it is efficientto interact with larger customers. The total spillover effect isθsize+γsizeω= 0.0283.Leverage The firm effect (γleverage) is negative, which can be explainedby the idea that a leveraged firm has financial obligations, andthese obligations constrain the firm’s production possibility fron-tier. We find that the exogenous spillover effect, θleverage, is262.4. Productivity Spilloversapproximately equal to zero. This means that, if we hold con-stant a customer’s TFP, the customer’s financial leverage causesno spillover effects. Therefore, a customer’s financial leverage isonly significant because it negatively impacts the customer’s ownTFP. The total spillover effect is equal to θleverage + γleverageω =0 + (−0.0592) (0.6081) = −0.036.Age Our estimates from Column 3 show that the firm effect is in-versely U-shaped, and peaks at age 14. Jensen et al. (2001),Van Biesebroeck (2005) and Fernandes (2008) find an inverseU-shaped relationship between firm age and productivity. Ourresults show that, in addition, the spillover effect is inverse U-shaped (this effect peaks at age 24), i.e. it is efficient to interactwith customers that are neither too young nor too old. This re-sult can be explained as follows: when a firm interacts with veryyoung customers, these customers are inexperienced at handlingtheir operations. For example, they may have unstable orderingcycles, which can trigger a large bullwhip effect across the supplychain. As the customers age, there is a learning-by-doing effectthat mitigates these inefficiencies (Syverson 2011). But firms areless prone to deviate from their customary practices (to inno-vate) at an advanced age. Therefore, a firm that interacts withold customers may adopt, or be influenced by, the old-fashionedpractices of their partners.Inventory We show that both the firm effect and the spillover effect arepositive. This means that firms with higher inventory turnoverare more productive and, in addition, that firms benefit fromhaving customers with high inventory turnover. A causal rela-tionship between productivity growth and inventory reductionhas been previously observed across automobile supply chains inJapan (Lieberman and Demeester, 1999).Location Being located in the U.S. has a positive impact on TFP. But ifwe hold the partners’ productivity fixed, a firm’s productivityis positively affected when it has a larger proportion of foreigncustomers. Similar results are found by Keller and Yeaple (2009),who show that foreign enterprises provide positive productivityspillovers to U.S. firms. If we sum up the indirect and directspillover effects, we find that (overall) it is beneficial to interactwith customers located in the U.S. west coast. The spillover272.5. Extensions and Robustness Checkseffect of geography, however, is not statistically significant.Summary of ResultsOur results show that a firm’s productivity benefits when the firm interactswith customers that have: (i) high productivity levels; (ii) large size; (iii)age between seventeen and twenty; (iv) high inventory turnover and; (v) lowfinancial leverage. The effect of geography is small, and not very robust.We note three interesting results. First, firm age has two counteractingeffects: while firms indirectly benefit from interacting with larger customers(because they are more productive), they directly benefit from interactingwith smaller customers. We argue that this is because large firms exertmore influence over smaller customers. Second, inventory turnover causesboth direct and indirect spillover effects, but financial leverage only causesindirect spillovers. This hints at the fact that a firm’s productivity is moresusceptible to the operational conditions of its peers, than it is to theirfinancial conditions. Third, after analyzing the magnitude of the effects,we find that the endogenous effect is the largest one. In other words, theendogenous channel is the primary channel through which productivity spillsover - more so than any exogenous channel. This means that interacting witha productive firm is very beneficial, even if the firm has “counterproductive”traits (e.g. high financial leverage). It is less beneficial to interact withunproductive partners (even if they have favorable exogenous traits).142.5 Extensions and Robustness ChecksWe now study some extensions to gauge the robustness of our model. In§2.5.1, we estimate productivity spillovers between a firm and its suppliers.In § 2.5.2, we re-estimate our results by controlling for selection biases inthe formation of supply chains. We also perform various (minor) robustnesschecks, which are discussed in § last statement must be interpreted with caution, because “beneficial” charac-teristics are also associated with high productivity. An alternative way of interpretingthis result is the following: if a firm could change any aspect of its customers (holdingeverything else constant), increasing the customers’ productivity would yield the highestbenefits for the firm’s productivity.282.5. Extensions and Robustness Checks2.5.1 Extension 1: Supplier-Base AnalysisIn the preceding sections, we looked at the impact of customers on theirsupplier’s TFP. While it is true that we can invert these relationships toextract the firms’ supplier bases, the resulting data overreport small firmsas suppliers. This is because firms only report (in the 10-K filings) thosecustomers that represent a large portion of their revenue; on average thesecustomers are large. Large firms, on the other hand, are unlikely to reportsmall firms.To understand why this is problematic, consider the following stylized ex-ample. Suppose A is a small firm whose annual revenue is equal to $100,000,and 20% of this revenue is obtained from firm B. Firm A will thus reportfirm B in the data. Now, suppose that firm X is a large firm whose annualrevenue is equal to $100 million, and also that firm X obtains 5% of its annualrevenue from firm B. As a result, the data will not capture the relationshipbetween firm B and firm X, even though the total value of trade between Xand B is much larger than the value of trade between A and B.For example, Walmart appears as a major customer for up to 106 firmsin a single year, but each of these firms cover a tiny fraction of Walmart’ssupplier-base. This will likely give rise to a reporting bias when estimatingthe impact of suppliers on a firm’s TFP.Having acknowledged this limitation, we extend our study to considerthe influence of suppliers on their customers’ TFP. Although the resultspresented in this extension may yield interesting insights, they should beinterpreted with caution (due to the limitations expressed above).Econometric ModelWe define WS as the weighted interaction matrix of firm-supplier relation-ships. This matrix has typical elementwSij =cji·(salesji)∑Nj=1 cji·(salesji)if cji = 10 otherwisewhere, recall, salesji is the amount paid by firm i to supplier j. Based onthis definition, we analyze the following structural equation:TFP =β1 + Xγ + WSXθ + ω(WS)TFP + u (2.9)We identify equation (2.9) through the strategies described in § Extensions and Robustness ChecksNo. of suppliers (%) Avg. cost0 18,125 (66.21%) 01 5,920 (21.62%) 18.7%2 1,382 (5.05%) 10.1%3 612 (2.24%) 15.6%≥4 1,337 (4.8%) 17.9%Table 2.7: Summary statistics about the supplier-base of each firm.In Table 2.7, we summarize the data on the (observed) supplier-base.This table also reports the cost of all purchases made by a firm to its sup-plier base (as a percentage of the firm’s total purchases). To calculate thisitem we use the total cost of goods sold (COGS) from Compustat, and letCostSi =∑j salesjiCOGSi. As we can observe in the table, the average cost ofobserved suppliers is relatively small, especially when compared with thecustomer-base. While major customers represent (on average) 25.5% of thefirm’s annual revenue, the observed suppliers represent 17% of the firm’sannual purchases. We also note that the average supplier size is equal to1.57, while the average size of a reported customer is equal to 6.63 (size ismeasured by the logarithm of total assets). This implies that, on average, areported customer is four times larger than a reporting supplier. Also, whilethe customer bases include up to 9 “major” customers, the supplier basessometimes comprise more than 100 suppliers.EstimatesWe present our model estimates in Table 2.8: we present OLS estimates inColumn 1; in Column 2, we present the Lee estimates without controllingfor correlated fixed-effects and; in Column 3, we present the Lee estimateswith controls for correlated network fixed-effects. A Hausman test revealsthat the model in Column 3 is the most robust one.The Lee estimates show that the endogenous effect (ω) is positive. Evenso, this estimate is small and statistically insignificant in the fixed-effectsmodel. This could be due to the fact that the suppliers are small and,hence, do not represent a significant portion of the firm’s purchases (recallthe Walmart example above). The reported suppliers are thus unlikely torepresent a significant influence on the firm. The qualitative properties ofthe exogenous effects are very similar to the estimates derived through thecustomer-base analysis.302.5. Extensions and Robustness ChecksEffect Variable(1) (2) (3)OLS Lee LeeCoeff. t-stat Coeff. t-stat Coeff. t-statEndogenous effect Sup. TFP -0.3453*** (-8.03) 0.2930*** (2.83) 0.0027 (0.03)Age 0.0149*** (5.32) -0.0004 (-0.20) 0.0100*** (3.72)Age2 -0.0004*** (-7.33) 0.0001 (1.23) -0.0002*** (-2.96)Exogenous Size 0.0284*** (3.59) -0.0492*** (-2.90) -0.0279 (-1.61)spillover Leverage -0.0001 (-0.78) -0.0001 (-1.26) -0.0001 (-0.78)effects Inv. Turnover -0.0483 (-0.42) -0.6361*** (-5.91) -0.3201*** (-2.60)Inv. Turnover2 0.1656* (1.72) 0.3200*** (5.51) 0.3013*** (3.69)Region: West 0.2689*** (4.48) -0.3353*** (-3.37) -0.0680 (-0.65)Region: Midwest 0.0902* (1.82) -0.3481*** (-4.17) -0.1824** (-2.09)Region: South 0.0959** (2.09) -0.2521*** (-4.17) -0.1047 (-1.61)Region Northeast 0.0720* (1.68) -0.2967*** (-3.93) -0.1773** (-2.25)Age 0.0155*** (7.60) -0.0172*** (-10.64) 0.0157*** (14.91)Age2 -0.0005*** (-15.75) 0.0003*** (8.97) -0.0005*** (-23.65)Size 0.1445*** (41.87) 0.0385*** (8.58) 0.1412*** (61.63)Leverage 0.0332 (1.52) -0.1773*** (-7.44) 0.0376*** (2.63)Firm Inv. Turnover 0.8590*** (15.12) -0.2764*** (-5.50) 0.8317*** (21.00)effects Inv. Turnover2 -0.1159*** (-7.09) 0.0465*** (3.35) -0.1135*** (-16.11)Region: West 1.0255*** (31.14) 0.1105*** (6.93) 1.0239*** (79.31)Region: Midwest 0.8812*** (29.38) 0.0249 (1.60) 0.8903*** (61.50)Region: South 0.5997*** (30.56) 0.0599*** (5.29) 0.5966*** (50.33)Region Northeast 0.7789*** (31.75) 0.0633*** (4.85) 0.7794*** (67.22)Intercept 1.7683*** (61.53)Number of Observations 27,376 27,376 27,376R-Squared 0.8809 0.0205 0.8763Fixed Effects (Network level) Yes No Yes*** p<0.01, **p < 0.05,*p <0.1Table 2.8: Model estimates for equation (2.9)312.5. Extensions and Robustness Checks2.5.2 Extension 2: The Formation of Supply ChainsIn the preceding sections we assumed that the formation of supply chainsis an exogenous process. But in reality firms make strategic decisions aboutwhether to select suppliers that are large or small, proximate or distant, etc.The underlying selection process will give rise to a non-randomly selectedsample. We thus need to consider the possibility that spillover effects arecorrelated with the selection of supply chain partners. We refer to thisproblem as partner selection bias.15To explain why this is a problematic issue, consider the following exam-ple: assume firm j is a customer of firm i and suppose that when a firm islocated overseas (as opposed to the U.S.), this induces a positive shock onthe partner’s TFP. So if j relocates its production facilities from the U.S.to overseas, we expect the productivity of supplier i to receive a positiveshock. But we can also expect that firm j’s desire to purchase from firmi will change as a result of j′s relocation. For example, j will drop i as asupplier if the shipping costs increase significantly.Fortunately, most of the supply chain relationships in our data lingerthrough several years. Note that 95% of the firm-customer relationshipsthat are found in period t − 1 are also found during period t and, on aver-age, a reported relationship lasts for over five years. This is likely due tothe presence of long-term contracts, and implies that most of the observedrelationships are the result of past decisions.While the above condition drastically mitigates the impact of this prob-lem, it does not completely eliminate it. To address this issue we build astrategic network formation model that draws on Goldsmith-Pinkham andImbens (2013). This model allows us to derive consistent estimates in thepresence of partner selection biases.16This approach is analytically complex, and we explain it in the Appendix.Roughly speaking, we build a dynamic approach that considers two sequen-tial processes. In the first process, firms select their supply chain partnersbased on a number of factors. We estimate these factors. And then, condi-tional on the selection process, we estimate the determinants of productivity15A similar problem arises in sociological networks, where individuals have a tendencyto befriend people with similar attributes. This is known as homophily.16Note that this type of bias plagues (virtually) every study that investigates spillovereffects across supply chain networks (including the ones cited in this manuscript). Thisproblem has not been controlled in the past, mainly because there were no theoreticalmodels to address it. As Bramoullé (2013) notes, “Goldsmith-Pinkham and Imbens pro-pose one of the first convincing applications” to deal with this problem. For this reason,our attempt to control for partner selection bias has some limitations.322.5. Extensions and Robustness Checks(i.e. the firm effects and the spillover effects).To jointly estimate the factors that influence both processes, we useMonte-Carlo-Markov-Chain methods coupled with Bayesian estimation. Wereport details about our model, and the results in the Appendix.Our main take-aways from this extension are the following. First, wefind (from Table A.1 in the Appendix), that two firms are more likely to en-gage in a relationship if: (i) their industries are compatible;17 (ii) the firmswere already engaged in a relationship; (iii) the firms are in close geographicproximity and; (iv) the supplier has large size and age. Other factors, suchas inventory turnover and financial leverage, did not play a statistically sig-nificant influence on the selection of supply chain partners.Second, we find that there are mild partner selection biases that causethe endogenous effect to be slightly upward biased (see Table A.2 in theappendix). However, the magnitude of these biases is very small, and notnearly large enough to overturn our main conclusions. We also find that theexogenous effects are practically unaltered by the presence of these biases.This is likely due to the fact that supply chain relationships last for severalyears implying that, to a large extent, the selection of supply chain partnersis exogenous to our model. As a result of this test, we are confident that ourmain results are robust to the presence of partner selection biases.2.5.3 Other Robustness ChecksIn this section we test the robustness of our main findings by varying someconstructs from our main model: in §2.5.3, we vary model constructs of theproduction function; in §2.5.3, we vary the constructs of the model used toestimate spillover effects.Due to space constraints, we do not present the tables for the results thatare discussed below, but all tables are available from the authors. Also, notethat the robustness checks presented below do not comprise an exhaustivelist. We performed other (minor) robustness checks, which are not discussedin this section.17We create a proxy to measure the compatibility between two industries by looking atthe historical connectivity between every pair of industries. We find, for example, thatthe leather and the apparel industries are very compatible.332.5. Extensions and Robustness ChecksProductivity EstimationIndustry ClassificationProductivity is the relative efficiency ranking of a firm within the industry itoperates. Any productivity estimates are thus unavoidably sensitive to theclassification scheme used to estimate them. To avoid inconsistencies acrossstudies, the literature often uses the Standard Industry Classification (SIC)code. In this essay we follow this standardized approach by classifying firmsinto an industry according to their 3-digit SIC code.We decided against using a 2-digit classification scheme, because thisclassification is too coarse. For example, under a 2-digit SIC code, we wouldhave had to group together firms producing pickup trucks, and firms pro-ducing bicycles, into the same industry. A 4-digit SIC code, on the otherhand, would have forced us to granulate firms into very specific industries.For example, we would have had to generate a separate industry for firmsthat manufacture leather luggage (SIC code 3161), and one for firms thatmanufacture leather purses (SIC code 3171).To ensure that our main conclusions are not sensitive to the industry clas-sification, we re-estimated our results by using a 2-digit and a 4-digit SICcode. We also used a 3-digit North American Industry Classification Sys-tem (NAICS) code. We find that our estimates are very robust to industryclassification. This model construct does not influence our main results.Intermediate MaterialsThe classic Cobb-Douglas production function, Y= AKβKLβL , has two in-put choices: capital (K) and labour (L). But it is not uncommon to finddifferent specifications for this function. A very popular alternative includes,in addition to labour and capital, a third input that measures intermediatematerials (M). Under this specification the Cobb-Douglas function becomesY = AKβKLβLMβM .There is no inherent advantage with either functional form, and ourchoice was motivated by the fact that the related literature uses a two-factor functional form (e.g. Imrohoroglu and Tuzel, 2014; Kellogg, 2011;Van Biesebroeck, 2005, etc.).We re-estimated our results by using a three-input production function,and found that the estimated input elasticities are βˆK = 0.3681, βˆL = 0.4697and βˆM = 0.2258. We used the resulting estimates to perform a visualcomparison of the firm’s productivity under both types of functions. Therelative ranking of firms does not seem to be affected by the choice of a342.5. Extensions and Robustness Checks2-input or a 3-input production function. In other words, productive firmslook productive (and unproductive firms look unproductive) regardless ofthe specification used. In the actual estimation of spillover effects, none ofthe results change their qualitative properties.Estimation of Spillover EffectsDynamic Spillover EffectsThe endogenous spillover effect measures how a firm’s productivity is af-fected by the productivity of its partners. As such, ω captures the effectof communication and collaboration with other firms, but also the result oflearning and mentoring.An important issue here is that learning effects are not always internalizedby the firm immediately but, rather, they diffuse through time. In otherwords, it may take two to three years for a firm to learn and adopt theproductive practices of their partners. But our model does not capturedynamic effects.We tested the robustness of our model by studying dynamic spillovers.In this extension, we assume that a firm’s productivity affects its partners’productivity with a lag. We study the following modelTFPt = β1 + Xtγ + WtXtθ +N∑n=0ωt−n(Wt−n)TFPt−n + utWe use the above model to study the lagged effect of the endogenous chan-nel. Note that the reflection problem does not affect the lagged variables.18Therefore, these lagged effects can be treated as exogenous regressors.We regressed various specifications of the above model by setting N equalto 1, 2 and 3, and also by excluding the spillover effect at time t. We didfind the presence of lagged effects. For example, by running a model withtwo lagged variables (i.e. by setting N = 2), we found that the endogenousspillovers were equal to ωt = 0.2619, ωt−1 = 0.07527 and ωt−2 = 0.118218.19All of these coefficients are statistically significant at the 1% level. If we dropthe lagged effects, our regression (for this same sample) shows that ωt =0.3026. All of our results are qualitative identical under both regressions.Our main conclusion is that our qualitative results are robust to theintroduction of lagged endogenous effects. However, as we see above, the18For a discussion about this issue, see Aakvik et al. (2013).19Note that this sample includes only those firms that appear for three consecutiveyears. The sample size is equal to 11,645.352.5. Extensions and Robustness Checkslearning effects that diffuse through time are non-trivial. These effects shouldbe analyzed in a more careful manner, in a separate study.Small NetworksA significant number of networks contain less than five firms. To makesure that our estimates are not affected by the inclusion of small networks(particularly when correcting for network fixed effects), we re-estimated themby excluding small networks. The results are not altered by the exclusion orinclusion of these networks.Variable DefinitionsWe relied on standard accounting definitions to define the vector of exogenouscharacteristics, Xi. Unfortunately there are multiple (valid) definitions forsome characteristics. In these cases we adopted the most commonly useddefinition (in related literature), or the choice was data-driven.To ensure that our main results are robust to the choice of definition, were-estimate these results by using alternative definitions:Inventory Turnover we defined inventory turnover as the ratio of netsales to inventory. However, it is more precise to define this variable as theratio of the Cost of Goods Sold to the average of inventory.Our choice was motivated by the fact that more observations report dataon annual sales than they do on the cost of goods sold. This allowed us todrop fewer variables. Note that this is not a big issue, given that there isa 97.4% correlation between both definitions. In the actual estimation, ourresults were unaltered by the choice of definition.Financial Leverage To measure financial leverage, we used the ratio oftotal debt (short plus long-term debt) to total assets. But in some stud-ies leverage only includes long-term debt in the numerator (e.g. Cardella,2013; Imrohoroglu and Tuzel, 2014). In other studies, leverage is measuredby using ratio of book value of total assets to the book value of equity (e.g.Patatoukas, 2011). We re-estimated our results using these alternative defi-nitions. The quantitative results are not altered by either definition.Geographic Location We could control for geographic effects by dividingfirms into states, or into one of the nine Census-Bureau designated divisions.We found that using the official census regions (West, Mid-West, South,362.6. ConclusionsNorth East) facilitates the exposition of our analysis, as opposed to dividingfirms into states, and analyzing the separate impact of each dummy variable.If we re-estimate our results by classifying firms into their home states,our results are qualitatively identical. As an aside note, it worth noting thatstates like California, New York and New Jersey have the most productivefirms (on average).2.6 ConclusionsWe provide new evidence about the link between Total Factor Productivityand supply chains. Our main contribution is to identify the various channelsthrough which productivity can spill over across firms. We identify the im-pact of a firm’s productivity on the productivity of its supply chain partners(the endogenous channel), and also the impact of the firm’s characteristicson these partners (the exogenous channels).Our estimates yield several interesting results. Among other things, wefind that interacting with productive firms (i.e. the endogenous channel) isthe largest source of productivity spillovers along the supply chain, more sothan any exogenous channel. We also find that a firm’s productivity is moresusceptible to the operational, than to the financial characteristics of thepartners.To arrive at the above results, we first had to deal with the reflectionproblem. This is often considered one of the most challenging economet-ric tasks. To this end, we used a novel econometric approach that exploitspartially-overlapping network interactions. Note that without controlling forthis problem, we would have found the opposite result, i.e. that interactingwith productive partners hinders a firm’s productivity (compare the esti-mates of the endogenous effect, ω, in Column 1 with ω in Column 3 - inTable 2.6).Identifying the joint impact of these two types of channels is more than aninteresting econometric exercise. Our estimates provide a precise descriptionabout how firms affect the productivity of their partners. These details canbe useful for practitioners at the time of building (and managing) supplychain relationships. To our knowledge, such estimates do not exist in theliterature, perhaps because of the identification challenges involved.Thanks to the recent infusion of data about inter-firm relationships, thereare several avenues for further research. For example, our model did nottackle the issue of spillover effects across competitors. The influence of com-petitors can be modeled by using the approach adopted by Bloom et al.372.6. Conclusions(2013). Another topic of research is to explore the impact of second- andthird-tier suppliers (and customers) on the productivity of the firm. It wouldalso be interesting to study the dynamic nature of the spillover effects, i.e.how the influence of partners persists over time. This essay contributesby providing a robust framework to estimate spillover effects across supplychains.38Chapter 3The Strategic Role of BusinessInsurance in Managing SupplyChain Risk3.1 IntroductionFirms routinely face the possibility of operational failures. For example, acontaminated input may cause a product recall, an industrial accident mayshut down production or a toxic spill may cause an environmental hazard.These events often lead to serious financial consequences that can threatenthe survival of the firm. The likelihood of a failure, however, can be mitigatedthrough costly effort such as operational maintenance or quality control.In a supply chain, preventing a failure involves coordinating the effortsof multiple firms. As a case in point, consider oil drilling operations. Thelikelihood of an oil spill depends both on the care taken by the driller andthe oil well cementer. The efforts of the two firms are partially substitutablebecause, for example, the oil driller can increase its drilling care to compen-sate for a poorly cemented oil well. However, the safety actions taken by thedriller are more effective when the well is appropriately built and maintainedby the cementing provider (and vice versa).Coordinating reliability efforts is complicated by two factors. First, theefforts of the firms may not be observable (e.g. the level of care taken byan operator in maintaining its equipment). This will give rise to moralhazard problems. And second, it may be impossible to identify the root-causeof the operational failure, which will lead to ambiguity about the degreeof responsibility of each party. This is a frequent problem in operationsinvolving “interdependent” systems (Kim and Tomlin, 2013). For example,an executive from EcoMotors argues that in the event of a product defectleading to a recall, “assigning responsibility for warranties gets messy. Was itjust a part that was not designed properly? Was it the environment that thepart was in, which typically is controlled by the automaker?” (Armstrong,393.1. Introduction2003). In other cases the failure automatically destroys all evidence aboutthe root-cause, e.g. an explosion leading to a major fire (Okes, 2009).Firms can deal with the above two problems through contractual clausesby allocating the financial burden of a failure, i.e. the ex post liability, tothose parties best positioned to prevent it. These clauses take the formof performance penalties, liability-sharing agreements or quality warranties,and can achieve the right incentives for effort provision. Contractual toolsare often used to apportion the potential costs of various types of failures,including product recalls arising from quality defects (Chao et al., 2009),oil spills (Hewitt, 2008) and the accidental releases of other toxic materials(Gallagher, 2012).But contractual agreements are not the only tool used by firms to allocatethe financial liability of an operational failure. Firms also have the abilityto transfer their liability away from the supply chain to third parties. Forexample, a firm can purchase business insurance to provide coverage for lossesarising from an operational failure. As a result, firms have two mechanisms todeal with failures: (i) contractual incentives, which allocate financial liabilitywithin the supply chain and; (ii) insurance coverage, which transfers theliability away from the supply chain.Most firms rely on some form of insurance coverage as part of their overalloperational strategy. In some cases, insurance offers coverage for losses aris-ing from uncontrollable factors, e.g. a natural disaster or a terrorist attack.But in many other cases insurance offers coverage for events where eitherthe operator or the supplier, or both, can potentially affect the likelihoodof the outcome. For example, in the U.S. most insurance companies offerservices like equipment breakdown insurance or boiler and machinery insur-ance to cover small losses arising from equipment failures. They also offerservices like product recall insurance, nuclear liability insurance or oil-spillinsurance to cover losses arising from more serious events. These servicescover a variety of industries including light & heavy manufacturing, utilities,steel machinery, mining & minerals, chemical products, hi-tech companies,etc. (RSA group, 2011).When the firms can affect the likelihood of a failure, the use of insurancemay lead to inefficiencies due to moral hazard. This is because insurancedecreases overall incentives to ensure operational reliability, by leading tothe “de-responsabilization of parties or agents in the supply chain” (Koganand Tapiero, 2007). Why then do firms buy insurance coverage for thesetypes of events? Would it not be more efficient to allocate all liability withinthe supply chain, to those firms who are in the best position to prevent anoperational failure?403.1. IntroductionOne answer to the above question is that insurance coverage serves aliquidity-enabling role, by allowing wealth-constrained firms to avoid thepossibility of bankruptcy or illiquidity due to an event causing large losses.For example, producers of nuclear energy can exchange the prohibitivelylarge costs associated with a nuclear meltdown for a manageable insurancepremium. Therefore, if all firms in a supply chain are wealth-constrained andthe overall wealth of all firms cannot cover the potential impact of a failure,they may need to seek third-party insurance. Insurance can play a secondrole in the presence of risk aversion, which is to allow risk-averse firms totransfer risk to a (risk-neutral) insurer. In other words, insurance improvesthe allocative efficiency of risk within the economy.But could the above two roles entirely explain the use of business in-surance in a supply chain? In this essay we show that, even if the supplychain has adequate wealth and firms are risk-neutral (i.e. in the absence ofthe above two roles), insurance may serve a purely strategic role within thesupply chain. Specifically, we show that without insurance some firms inthe supply chain may excessively free-ride on the efforts of other firms. Thepurchase of insurance can serve as a mechanism that allows a firm to credi-bly commit not to increase its effort and, thereby, to mitigate the free-ridingproblem. In other words, some firms in the supply chain can strategicallyuse insurance as a commitment mechanism.To see the strategic role played by insurance, consider a supply chainrelationship where one of the firms has sufficiently large wealth to cover anypotential losses arising from an operational failure, but the other firm hassevere wealth constraints. In this case contractual agreements alone mis-allocate incentives along the supply chain. Specifically, contractual toolsalone leave the wealth-constrained firm with inefficiently low incentives toexert effort because its is unable to take on the appropriate level of liabil-ity. Conversely, the “wealthy” firm ends up with excessively high incentives.Because effort is substitutable, any increase in the effort of the wealthy firmwill further undercut the incentives of the wealth-constrained firm to ex-ert effort. Insurance can play a strategic role by allowing the wealthy firmto credibly commit not to exert effort beyond some level and, in turn, thewealth-constrained firm has less incentive to free-ride. Insurance can there-fore mitigate the distortion in effort provision and improve total welfare.The above results are particularly relevant to supply chains character-ized by an uneven wealth distribution. These types of relationships are be-coming increasingly common as “larger corporations are looking to partnerwith small, specialized companies” (Business Development Bank of Canada,2013). Also, in emerging economies it is not rare to see large, wealthy multi-413.2. Literature Reviewnationals partnering with small and medium enterprises (Etemad et al.,2001). In these cases, our results imply that the uneven distribution ofwealth would lead to firms being more likely to buy insurance as a strategictool. In general, our results imply that insurance coverage and contractualincentives are not necessarily substitutes, but may complement each other.3.2 Literature ReviewThis essay bridges two research streams: (i) supply chain contracts and; (ii)risk management and insurance.Supply Chain Contracts: We are interested in the sub-stream thatfocuses on contracts coordinating reliability investments. Kim et al. (2007)and Chu and Sappington (2010) are prominent examples of this literature.Specifically, we consider a context where the operational outcome can beinfluenced by the efforts of both the operator and the supplier, and also thatthese efforts are unobservable. This is a setting characterized by double-sidedmoral hazard, which is also studied by Roels et al. (2010) and Jain et al.(2013b). Second, our model assumes that the root-cause of an operationalfailure cannot always be attributed to either party; similar assumptions aremade by Saouma (2008) and Kim and Tomlin (2013).Our model considers the case where firms have limited wealth in aprincipal-agent setting. This literature stems from Sappington (1983), whoargues that “contracts in which the liability of one or more parties is ex-plicitly limited are very common in practice.” In the economics literature,this topic has been extensively explored; some of the most notable exam-ples are Innes (1990), Holmstrom and Tirole (1997), Oyer (2000), Gromband Martimort (2007) and Poblete and Spulber (2012). Some examplesfrom the management literature include: Saouma (2008), who studies out-sourcing relationships and warranties in a setting where the suppliers havelimited wealth; DeVéricourt and Gromb (2014), who study capacity invest-ments under limited liability and; Desiraju (2004), who studies intrabrandcompetition under the same assumption.Risk Management and Insurance: The main goal of this essay isto contribute to a better understanding of the interaction between businessinsurance and supply chains contracting. This literature is limited. In theoperations management literature, Dong and Tomlin (2012) study the inter-play between business insurance and inventory management, and find thatinsurance can increase the marginal value of inventory and the overall valueof emergency sourcing. Unlike this essay, Dong and Tomlin consider a model423.3. Model Preliminarieswhere firms obtain coverage for uncontrollable events, e.g. natural disastersor terrorist attacks. Therefore, the authors do not need to consider issuesrelated to moral hazard, which are at the core of our model.Unlike the operations management literature, there is an extensive liter-ature in economics that explores the optimality of insurance in the presenceof moral hazard. Winter (2000) summarizes this literature. Within this largeliterature, our model is most closely related to Tommasi and Weinschelbaum(2007) who study the robustness of principal-agent contracts to the intro-duction of third-party insurance. Our results differ from this paper in twokey respects. First, while their analysis is driven by the assumption of riskaversion and single-sided moral hazard, we assume that the parties are risk-neutral and subject to double-sided moral hazard. Second, in their paperinsurance opportunities are available for the agent, not for the principal; weassume that the principal has the ability to purchase insurance coverage.Due to this reason, they find that insurance decreases welfare while we showthat insurance can improve total welfare in contractual relationships.3.3 Model Preliminaries3.3.1 Operational FeaturesA risk-neutral operator (O) receives revenue pi from operating a system thatrequires the technical expertise of a risk-neutral supplier (S). Consider thefollowing examples: (i) an oil driller that delegates all cementing operationsto an oil well cementer or; (ii) an equipment operator that outsources allsupervision and maintenance tasks to a service supplier. The system is sub-ject to unexpected operational failures, e.g. a biohazard spill leading to anenvironmental accident, or a defective product that must be recalled. Thesefailures lead to financial losses for the operator from property damages, clean-up costs, production interruption, etc. Let X ∈ {0, x} be a random variablerepresenting the “failure costs” borne by the operator. If X = 0, operationshave performed as planned, and the operator incurs no losses. If X = x > 0,an operational failure has occurred, and the operator incurs losses equallingthis amount.20Failure Probability and Reliability Efforts:To diminish the likelihood of a failure, the supplier and the operator canexert costly but unobservable effort. For example, the supplier may use20In §3.5 we consider an extension where the failure costs have a continuous support.433.3. Model Preliminariesbetter quality inputs or improve the design of its product or service. Theoperator can increase the level of care when performing operations, minimizethe systems’ exposure to strenuous conditions, hire skilled personnel, etc.Let eS ≥ 0 and eO ≥ 0 denote the efforts of the supplier and the op-erator, respectively, where eS and eO are the dollar investments to improveoperational reliability; e∗S and e∗O denote the optimal effort levels.The probability of an operational failure is F (eS , eO) = 1(1+eO)β(1+eS)1−β,where parameter β ∈ (0, 1) represents the sensitivity of the failure proba-bility to the operator’s effort (relative to the supplier’s). When β is lowreliability is more sensitive to the supplier’s effort and less sensitive to theoperator’s effort, and vice versa. Note that F (eS , eO) is an inverse Cobb-Douglas function, which reflects the idea that the efforts of the operatorand the supplier are substitutable and also collaborative in nature. Effortis substitutable because the operator can exert higher effort to compensatefor situations where the supplier exerts lower effort, and vice versa. How-ever, it is collaborative because the effort of one firm is enhanced when theother firm exerts high effort, i.e.∂2F (eS ,eO)∂eS∂eO> 0. Cobb-Douglas functions arecommonly assumed in papers involving collaborative effort in supply chains,including Roels et al. (2010) and Kim and Netessine (2013).Failure Attribution:We focus on settings where the root-cause of an operational failure cannotbe identified. This assumption is often the norm in practice and is studiedin a subfield of reliability analysis called dependent failure analysis. Fromthis literature we have identified three settings where this assumption holdsand, hence, where our model is applicable:1. Cascading Failures: Large-scale operations often require the inputof interdependent subsystems (e.g. power generation plants, oil and gaspumping, etc.), some of which need the managerial expertise of specializedsuppliers. When there is a high level of interdependency, the subsystemsare highly susceptible to experiencing a cascading failure, where the failureof one subsystem triggers the subsequent failure of other subsystems (Er-icson, 2005). In these cases it is either impossible, or prohibitively costlyto, to identify the subsystem responsible for the failure. Kim and Tomlin(2013) study liability allocation rules across systems that are susceptible toexperiencing cascading failures.2. Destructive Failures: In certain operational failures, evidence is dam-aged or destroyed following the event and, as Okes (2009) explains, this oftenmakes it impossible to determine the root-cause of the problem. For exam-443.3. Model Preliminariesple, following the explosion of a large wind turbine in Androssan, UK, “muchof the evidence was burned, and Infinis [the wind farm operator] and Vestas[the supplier of wind turbines] disagree on which was the key initial cause ofthe destructive fire” (New Scientist, 2013).3. Commingled and Homogeneous Goods: Whenever the efforts of themultiple firms contribute to the production of a good that is commingled,non-modular or homogeneous (e.g. chemical substances, food products), itis a challenge to determine the degree of responsibility of each party forthe operational failure. For example, in 2007 a tainted food incident fromConAgra caused approximately 15,000 people in the U.S. to fall sick, but“ConAgra could not pinpoint which of the more than 25 ingredients in its pieswas carrying salmonella” (New York Times 2009).21 Also, in July 2013 an oil-cargo train derailment caused a massive explosion in Lac-Megantic, Quebec.To date, the authorities have not been able to determine if the explosion wascaused by chemical contaminants in the oil (from a previous shipment), orbecause the oil itself contained high levels of flammable hydrogen sulphidegas (The Globe and Mail 2013).3.3.2 ContractsIn the face of a potential failure, the operator needs to optimally apportionthe liability for the accident. The operator can apportion some of the po-tential losses upstream to the supplier, through a contractual agreement. Inaddition, the operator can purchase insurance coverage to allocate some ofthe burden away from the supply chain to a third party insurer. We explainbelow the specifics of both agreements, (i) the procurement contract and (ii)the insurance policy.Procurement Contract:The operator procures the services of the supplier by offering a ‘take-it-or-leave-it’ contract, T (w, y,X). This contract consists of a fixed-fee payment,w ≥ 0, and a liability-sharing (or penalty) rate, y ∈ [0, 1]. The penalty ratesplits the cost of the failure between the contracting parties: the supplier paysyX, and the operator pays the remainder. The cash flows of the contract(from the operator to the supplier) are T (w, y,X) = w − yX, and the21More generally, the FDA (2010) states that whenever there is a contaminated foodproduct “traceability has proved particularly difficult because of the complexity of thedistribution system and the practice within pack houses supplying the US market of co-mingling produce.”453.3. Model Preliminariesexpected transfer payment is EX [T (w, y,X) |eS , eO] = w − yE [X|eS , eO],where E [X|eS , eO] = xF (eS , eO).This contractual form is widely observed in practice, where the suppliergets a fixed payment but a proportion of any liability (contingent on therealization of performance) is subtracted (Kim et al., 2007). By reviewingdifferent procurement agreements, we found numerous examples of contractsthat adopt this apportioning mechanism. For example, in a distributionagreement signed in 2001 between the technology companies Lucent Tech-nologies and Agere Systems, the parties agreed to split any costs arising fromproduct defects and accidental releases of contaminants; Lucent agreed tobe liable for 86% of the costs, and Agere for the remaining share (article VI,clause i).Note that sometimes the parties will first agree to do a preliminary inves-tigation to determine the root-cause of the failure, and the liability-sharingclause will only apply if the investigation is inconclusive.22 While this behav-ior is certainly observed, we assume that the root-cause of a failure cannotbe identified. But this assumption is not too restrictive because, as Moslelhet al. (1998) put it, “all too often investigations of failure occurrences... donot determine the root causes of failures.”Insurance Contract:The operator has access to an insurance market to cover any losses arisingfrom the operational failure.23 The operator will thus negotiate an insurancepolicy with a third-party insurer. To model this relationship we reviewed nu-merous insurance policies and spoke with practitioners. We identified threekey components (schedules) that characterize these policies: (i) a scheduleof insured events and losses; (ii) a payments schedule and; (iii) a scheduleof excepted causes.1. Schedule of Insured Events and Losses: This schedule defines thoseevents for which the operator can obtain coverage (e.g. an oil spill or aproduct recall), and the types of losses covered by the policy. In our model22For example, in a chemical manufacturing contract between APP Pharmaceuticals andNew Abraxis (2007, clause 12.9), the parties agreed to share the costs of a product recallonly if the fault of the recall cannot be determined through preliminary tests performedby an independent FDA testing agency.23Throughout the base model we assume that the operator has full control over the levelsof insurance that are purchased in the supply chain. In reality, however, the supplier mayhave the option to purchase insurance to cover some of its losses. We analyze this case inSection Model Preliminarieswe assume that both the operational failure and the failure costs are definedwithin this schedule.242. Payment Schedule: This schedule includes the cash flows of the policy.We consider an arrangement where the operator pays an insurance premiumand, in exchange, obtains a coverage reimbursement. Let v ∈ [0, 1] denote thecoverage level, which is equal to the proportion of the failure costs reimbursedby the insurer in the event of an operational failure. This reimbursementamounts to vx.The insurance premium, P , is an upfront payment made by the operatorto the insurer. In practice, the premium is quoted as the sum of three factors:(i) the actuarially fair premium, (ii) an additive load, and; (iii) a proportionalload. The actuarially fair premium represents the expected reimbursementto the operator, the additive load accounts for the fixed transaction costsof providing insurance and the proportional load accounts for costs that areproportional to the amount of coverage. Therefore, the premium is equal toP (v|e∗S , e∗O) = vE [X|e∗S , e∗O]︸ ︷︷ ︸actuarially fair premium+ ladd︸︷︷︸additive load(3.1)+ lpropvE [X|e∗S , e∗O]︸ ︷︷ ︸proportional loadThis type of pricing scheme has also been modeled in the literature, forexample, by Patel et al. (2005) and Dong and Tomlin (2012). The cashflows of the insurance contract (from the operator to the insurance provider)are equal to I (v, P,X) ≡ P − vX, and the expected cash flows areEX [I (v, P,X) |eS , eO] = P − vE [X|eS , eO] .We make the following three assumptions. First, we assume that theinsurer is knowledgeable about the product technology, i.e. the insurer isinformed about the function E [X|eS , eO], but cannot observe the effortprovision of each firm. Second, we assume that the insurer can observethe procurement contract, (w, y), before setting P . Therefore, the insur-ance provider can infer the incentive compatible levels of effort, e∗S and e∗O.Both assumptions are standard in the insurance literature (see Winter 2000).Third, to focus on the intuition behind our results we let ladd = lprop = 0,i.e. the insurance premium is equal to the actuarially fair level of coverage.24For example, in a product recall insurance policy from AIG Insurance Ltd. (2010), thepolicy defines an insured event as “any Product Recall or Government Recall resulting fromany: (a) Defect; (b) Malicious Product Tampering; or (c) Product Extortion.” The samepolicy defines the insured losses as “all reasonable and necessary (a) Insured’s ProductRecall Costs; (b) Business Interruption Costs; (c) Replacement Costs. . . ”473.3. Model PreliminariesOur results do not depend on this assumption because both loads reflecttransaction costs associated with purchasing insurance and, therefore, theonly influence that these factors play in our model is to (uniformly) shift theresults without adding any intuition.Finally, note that the payment schedule of a policy generally includesmore complicated payment arrangements, including deductible levels. In§3.5.3 we show that our results are robust to these complications.3. Schedule of Excepted Causes: An insurance policy also defines aset of excepted causes for which the insurer is exempted from covering theinsured. These exceptions generally include unacceptable actions from theinsured party, or matters uninsured under the law.25 We assume that theparties will not engage in any behavior that might lead to an excepted causeof a failure, e.g. unreasonable negligence, violations of the law, etc.A possible misconception is that the insured party will not be able toobtain coverage unless the root-cause of the failure can be clearly identified.Note that under the lawThe burden is on an insured to establish that the occurrence formingthe basis of its claim is within the basic scope of insurance coverage.And, once an insured has made this showing, the burden is on theinsurer to prove the claim is specifically excluded (Supreme Courtof California, 1989).This means that to obtain coverage the policy holder only needs to showthe existence of insured losses arising from an insured event. The insurercan only be exempted from making a reimbursement if it can convincinglyshow that the cause is excepted in the contract (e.g. if the insurer can showthat the operator was unreasonably negligent, intentionally tampered withthe product, etc.). And, as mentioned earlier, we abstract away from theseissues.3.3.3 Model DynamicsIn Figure 3.1 we illustrate the timeline of this model. In stage one the op-erator designs the procurement contract, T (w, y,X), and chooses the levelof insurance coverage, v. Next, the supplier accepts or rejects the deal of-fered by the operator. If the supplier accepts the arrangement, the operatorpurchases the insurance policy and makes the ex ante payments (i.e. theinsurance premium and the fixed fee, P and w).25In the Product Recall Insurance policy mentioned in Footnote 24, AIG defines a num-ber of excepted causes, including “Uninsurable matters under the law”, “Asbestos”, “In-tentional violation by the Insured of any governmental or regulatory requirements”, etc.483.3. Model PreliminariesFigure 3.1: Model timelineOperations begin in stage two. In this stage, the operator and the sup-plier exert effort levels eO and eS . Note that because we assume that theefforts of both parties are unobservable, the sequence in which effort is ex-erted will not affect the results analyzed below.26After both parties have exerted effort, the operational performance, X ∈{0, x}, is realized. The cash flows, yX and vX, are made at the end of theoperational period.3.3.4 Payoffs and Wealth ConstraintsPayoff FunctionsThe supplier’s and the operator’s ex post profits areΠS (eS , w, y,X) = T (w, y,X)− eSΠO (eO, w, y, v, P,X) = pi −X − T (w, y,X)− I (v, P,X)− eO26This is because, according to the Principle of Interchange of Moves, the order of playis “immaterial if one player does not have any information about the other player’s actionwhen making his choice.” (Osborne and Rubinstein, 1994). In Section 3.5.1, however, westudy a sequential game where the supplier is the first party to exert effort, and this effortis observable. Our results are robust to this setting.493.3. Model PreliminariesTherefore, the expected profits for both firms areEX [ΠS (eS , w, y,X) |eO] = EX [T (w, y,X) |eS , eO]− eSEX [ΠO (eO, w, y, v, P,X) |eS ] = pi − EX [X|eS , eO]− EX [T (w, y,X) |eS , eO]−EX [I (v, P,X) |eS , eO]− eO.Wealth Constraints:We focus on scenarios characterized by low-probability but high-impact fail-ures (e.g. a product recall, a biohazard accident, etc.). In this context firmsare often unable to sustain large financial losses. We assume that at theoutset (i.e. prior to any contractual agreement) the operator has wealthWO ≥ 0 and the supplier has wealth WS ≥ 0. Moreover, we assume thatboth parties will only consider contracts where they are guaranteed to endwith non-negative wealth at the end of stage two. Therefore, the followingconditions must be satisfiedWO + ΠO (eO, w, y, v, P, x) ≥ 0 (WCO)andWS + ΠS (eS , w, y, x) ≥ 0 (WCS)where, ΠO (eO, w, y, v, P, x) and ΠS (eS , w, y, x) refer to the ex-post profitsof the firms in the event of an operational failure (i.e. when X = x).Firms generally write liability limits as a condition to enter into any con-tractual agreement; these constraints have been thoroughly studied in theliterature (see §2). As Sappington (1983) explains, these clauses can be es-tablished as bankruptcy or insolvency provisions, where no further liabilitycan be imposed on the firm if it reaches a state of insolvency. The liabil-ity limits can also be explicitly stated as fixed dollar amounts.27 Saouma(2008) also uses these constraints to study outsourcing relationships, by ar-guing that “excessive liability resulting from a single faulty product can driveeven large suppliers into bankruptcy.” Similarly, Desiraju (2004) argues thata principal reason these clauses arise is due to “equity considerations thatmandate the guarantee of an appropriate level of well-being” for the variousparties involved in a contractual relationship.27The following is a sample limited liability clause drawn from a contract for the Manu-facturing of pharmaceutical inputs: “Notwithstanding anything herein to the contrary, inno event will the GENERICO Indemnified Parties have any liability to NEW ALPHA orany of its Affiliates, or to any third party in connection with this Agreement, for monetaryDamages in excess of $100 million in the aggregate” (Manufacturing Agreement betweenNew Abraxis Inc. (New Alpha) and APP Pharmaceuticals, LLC. (Generico), November,2007; Clause 12.9).503.4. Analysis3.4 AnalysisIn this section we study the interaction between liability-sharing throughcontracts and liability-sharing through insurance. We present a benchmarkscenario in §3.4.1, by assuming that the operator and supplier are organizedas a centralized entity. In §3.4.2 we consider the case where the firms operateas decentralized entities.3.4.1 Benchmark Scenario: Centralized Supply ChainWhen the operator and the supplier are organized as a centralized firm, acontract between these parties is unnecessary. The centralized supply chainthus finds the optimal level of effort and insurance coverage to maximize thetotal expected profits, E [Π] (where Π = ΠO + ΠS). This supply chain haswealth W = WO + WS and is unable to bear losses beyond this threshold.To find the optimal strategy for the centralized chain, we solve our modelthrough backward induction.Stage 2: Optimal Effort Levels:In stage 2, the centralized chain jointly chooses effort levels to maximizeEX [Π (eS , eO, v, P,X)] = pi − ((1− v)EX [X|eS , eO] + P + eS + eO)The level of insurance coverage, v, and the insurance premium, P , havealready been determined in stage 1. Note that since the premium is a fixedtransfer to the insurer made in stage 1, the optimal levels of eS and eO donot depend on the premium P .28Lemma 1. The centralized supply chain sets effort levels e∗O = eFBO ande∗S = eFBS satisfyingeFBO (v) =√x (1− v)β2−β(1− β)1−β− 1 and eFBS (v) =√x(1−v)(1−β)1+βββ − 1 (3.2)Proof: All proofs are found in the appendix. According to Lemma 1, the centralized supply chain sets effort levelssuch that these levels are proportional to the influence that each firm has in28When the failure costs, x, are small enough, then e∗O and e∗S are non-positive. In thesesituations, none of the parties exert effort and the model becomes trivial. For this reasonthroughout this essay we focus on the case where x is large enough so that e∗S > 0 ande∗O > 0.513.4. AnalysisFigure 3.2: First best level of insurance.mitigating the failure probability. To see this note thateFBS +1eFBO +1= 1−ββ , whereβ and 1 − β represent the relative sensitivities of the failure probability tothe operator’s and supplier’s efforts respectively. This ratio represents thefirst-best allocation of effort.Stage 1: Optimal Insurance Coverage:The centralized supply chain chooses the profit maximizing level of insurancecoverage, subject to the constraint that its ex post wealth must be non-negative. Recall that the insurance premium is priced in accordance withthe scheme specified in equation (3.1).Let ΠFB (v,X)≡Π(eFBS (v) , eFBO (v) , v, P(v|eFBS (v) , eFBO (v)), X)repre-sent the profit function evaluated at the optimal effort levels. Hence, in stage1 the centralized supply chain solvesmaxv∈[0,1]EX[ΠFB (v,X)]subject to ΠFB (v, x) +W ≥ 0. The following proposition characterizes thefirst best level of insurance coverage.Proposition 2. If W+pi ≥ x+√xββ(1−β)1−β−2, the centralized supply chainpurchases no insurance coverage, i.e. v∗ = 0. IfW+pi < x+√xββ(1−β)1−β−2,the centralized supply chain purchases insurance coverage v∗ = vFB > 0,where vFB solves√x(1−vFB)ββ(1−β)1−β= W + pi + 2− x(1− vFB).523.4. AnalysisWe illustrate this proposition in Figure 3.2. The centralized supply chainonly purchases insurance if, given the realization of an operational accident,the potential losses are large enough to exceed the entire wealth of the chain,i.e. if W + ΠFB (0, x) = W + pi − x−√xββ(1−β)1−β+ 2 < 0. In other words,in a centralized setting insurance only plays the liquidity-enabling role, thatis, to ensure the financial viability of the supply chain.3.4.2 Decentralized Supply ChainsWe now consider the case where the operator and supplier operate as de-centralized entities. As in §3.4.1, we solve this model through backwardinduction.Stage 2: Incentive Compatible Effort Levels:In stage 2 the operator and supplier independently set the profit-maximizinglevels of effort. The contract between the operator and supplier, (w, y), andthe insurance contract, (v,P ), have been determined in stage 1.The supplier and the operator have best response func-tions e˜S |eO ≡ arg maxeS≥0EX [ΠS (eS , w, y,X|eO)] and e˜O|eS ≡arg maxeO≥0EX [ΠB (eO, w, y, v, P,X|eS)]. We derive the Nash equi-librium effort levels in Lemma 3.Lemma 3. The operator and the supplier exert effort levels e∗Oand e∗S, wheree∗O (y, v) =√x (β (1− y − v))2−β((1− β) y)1−β− 1 and e∗S (y, v) =√x((1−β)y)1+β(β(1−y−v))β− 1The effort levels depend on the penalty rate and the level of insurancecoverage, y and v. This leads us to our next result, Lemma 4, which describesthe failure probability as a function of these parameters. This lemma helpsus simplify our analysis in subsequent sections.Lemma 4. Let Φ (y, v) ≡ F (e∗S (y, v) , e∗O (y, v)) denote the failure probabil-ity as a function of the penalty, y, and the level of insurance coverage, v. Wehave thatΦ (y, v) =[x (β (1− y − v))β ((1− β) y)1−β]−1533.4. AnalysisStage 1: Optimal ContractsIn this stage we derive the optimal operator-supplier contract, w and y, andthe optimal level of insurance, v. The operator’s problem is to solvemaxw≥0,y∈[0,1],v∈[0,1−y]EX [ΠO (e∗O, w, y, v, P (v|e∗S , e∗O) , X) |e∗S ]subject toEX [ΠS (e∗S , w, y,X) |e∗O] ≥ 0 (IR)ΠS (e∗S , w, y, x) +WS ≥ 0 (WCS)ΠO (e∗O, w, y, v, P (v|e∗S , e∗O) , x) +WO ≥ 0 (WCO)IR represents the individual rationality constraint for the supplier; WCSand WCO represent the wealth constraints for the supplier and for the op-erator. Note that the incentive compatibility constraints have already beenembedded in the problem, through e∗S = arg maxeS EX [ΠS (eS , w, y,X|e∗O)]and e∗O = arg maxeO EX [ΠO (eO, w, y, v, P,X|e∗S)]. To simplify our analysiswe first present the results for three special cases:• Special case UC - unconstrained wealth: we assume thatWO =∞and WS =∞.• Special case SC - Supplier with wealth constraints: we assumethat WO =∞ but WS <∞.• Special case OC - Operator with wealth constraints: we assumethat WO <∞ but WS =∞.After analyzing these three special cases we present the general results, whereWS ≤ ∞ and WO ≤ ∞.Special Case UC: Unconstrained WealthAssume that both parties are financially unconstrained, i.e. WS = WO =∞.In this case, the wealth constraints WCO and WCS are never binding. Wecharacterize the optimal contracts in Proposition 5.Proposition 5. When WO = ∞ and WS = ∞, the optimal contractingparameters are given by v∗ = vUC, y∗ = yUC and w∗ = wUC, where543.4. Analysis• vUC = 0• yUC ={0.5 if β = 0.51−β2−√β(1−β)(1+β)(2−β)1−2β if β 6= 0.5• wUC =xyUCΦ(yUC , 0)+ e∗S(yUC , 0)In the absence of wealth constraints the decentralized supply chain doesnot purchase insurance coverage, i.e. vUC = 0. This is because insuranceexternalizes liability away from the supply chain and, therefore, reduces theincentives of each firm to invest in reliability. As a result, the operator findsit more profitable for the supply chain to internalize all the financial liability,especially given that the supply chain has adequate wealth to self-insure.Since the failure probability depends both on the efforts of the operatorand supplier, the operator transfers some of the liability to the supplier, byoptimally choosing a penalty rate equal to yUC . As a result, the operatorinternalizes a proportion of the liability equal to 1−yUC , and the supplierinternalizes a proportion equal to yUC .Under this penalty the ratio of efforts is equaleUCS +1eUCO +1=√(1−ββ)3 (1+β2−β),where eUCO ≡ e∗O(yUC , vUC)and eUCS ≡ e∗S(yUC , vUC). The ratio is differentfrom the first best ratio (recall thateFBS +1eFBO +1= 1−ββ ), because the operator mustcoordinate the supply chain efforts by accounting for the double-sided moralhazard problem. For this reason, the penalty adjusts the effort levels so thatthe marginal effort of each firm is proportional to its influence in mitigatingthe failure probability. Observe that∂F(eUCS ,eUCO )/∂eS∂F(eUCS ,eUCO )/∂eO= 1−ββ(eUCO +1eUCS +1).29Special Case SC: Supplier with Wealth ConstraintsWe next look at the case where the supplier has finite wealth, but the op-erator has no wealth constraints, i.e. WS < ∞ and WO = ∞. This impliesthat constraint WCO is never binding.In Proposition 6 we present our main results for this case. This propo-sition shows the existence of two key wealth thresholds for the supplier,denoted by W IS and WIIS (where WIS > WIIS ). The threshold WIS is suchthat when the supplier’s wealth WS is greater than W IS , constraint WCS29This ratio is analogous to the one shown by Bhattacharyya and Lafontaine (1995) intheir seminal paper on the nature of share contracts under double-sided moral hazard.553.4. Analysisdoes not bind. When this happens, the operator’s problem reduces to theunconstrained case (UC). If on the other hand WS < W IS , the supplier’swealth constraint binds at optimum. This means that the supplier is unableto sustain large losses. Given this inability, the operator optimally reducesthe penalty rate that the supplier bears. The operator, however, only finds itoptimal to purchase insurance coverage when WS is less than the thresholdW IIS .Proposition 6. Assume that WS < ∞ and WO = ∞. Define W IS ≡yUCx(1− ΦUC)and W IIS ≡ x (1− β) (1− 2m), where m satisfiesm1−2m =1−β1−Φ(m,0) . The optimal contracting parameters are given by v∗ = vSC,y∗ = ySC and w∗ = wSC, where• vSC =0 if WS ≥W IS0 if WS ∈ [W IIS ,WIS)max{(1−β)(1−2ySC)2−β ,x(1−β)(1−2ySC)−WSx(1−β)+WS/ySC}if WS < W IIS• ySC =yUC if WS ≥W ISWSx(1−ΦSC) if WS ∈ [WIIS ,WIS)max{WSx(1−ΦSC) ,ΦSC(1−β)2−β(1+ΦSC)}if WS < W IIS• wSC =wUC if WS ≥W ISxySC + e∗S(ySC , 0)−WS if WS ∈ [W IIS ,WIS)xySC + e∗S(ySC , vSC)−WS if WS < W IISwhere ΦSC ≡ Φ(ySC , vSC).We illustrate the above results in Figure 3.3. When WS ≥ W IS , theoperator sets a penalty rate equal to yUC and purchases no insurance coverage(as in the unconstrained case). If WS < W IS the supplier is unable to sustainlarge financial losses and, therefore, the operator has to decrease the penaltyfrom yUC to ySC . When the operator decreases the penalty, it takes awayfrom the supplier a share of the failure costs. Note that this share is equalto(yUC − ySC)X.The operator now needs to decide whether to: (i) absorb this share,by retaining it or; (ii) transfer some of it away from the supply chain, bypurchasing insurance coverage. If WS ∈ [W IIS ,WIS), the operator choosesto retain this share, i.e. to internalize the liability. However, if the wealth563.4. AnalysisFigure 3.3: Optimal parameters - case SC, for β = 0.4, x = 100,pi = 45.constraint of the supplier is very stringent, i.e. whenWS < W IIS , the operatoroptimally transfers some of this share to the insurer by purchasing coverage.It may seem optimal for a risk-neutral and wealth-unconstrained operatorto always internalize the liability (instead of purchasing insurance). After all,when the operator internalizes the liability it has higher incentives to exerteffort. On the other hand, when the operator purchases insurance, theseincentives are reduced and the likelihood of an operational failure increases.So why does the operator purchase insurance when the supplier’s wealthconstraint is very severe? The intuition is as follows. Because the effortsof the operator and supplier are partially substitutable, any increase in theeffort of the operator has a negative externality on the incentives of thesupplier. In other words, when the operator chooses to keep the extra shareof the liability, the supplier knows that the operator has higher incentivesto increase effort (in stage 2). Therefore, the supplier has an incentive tofree-ride on the efforts of its counterpart and, as a result, it further reducesits own effort levels. This implies that the increase in the operator’s effort ispartially offset by a reduction in the effort of the supplier. In other words,when the operator internalizes the extra share of the liability, the supplychain is subject to an effort distortion.Consider instead what happens when the operator chooses to external-ize the failure costs through insurance. In this case the overall incentivesto exert effort are reduced. Hence, the operator needs to trade-off the effortdistortion caused by the free-riding problem with the reduction in the overallincentives to exert effort (caused by insurance). Our results show that whenthe supplier’s wealth constraint is not stringent, i.e. when WS ∈ [W IIS ,WIS),573.4. AnalysisFigure 3.4: Welfare parameters - case SCthe effort distortion in the supply chain is preferred to the dampening inefforts. In other words, it is better to internalize all liability. When the sup-plier’s wealth constraint is very stringent however, i.e. when WS is less thanthe thresholdW IIS , it is better to dampen the effort incentives by purchasinginsurance than to further distort the supply chain efforts (see Figure 3.3).The operator can achieve this by seeking insurance as a commitment not toincrease effort. This effectively reduces the supplier’s incentives to free-rideand, therefore, mitigates the effort distortion in the supply chain.Note that the operator may choose to buy insurance, not as a mechanismto ensure the financial viability of the supply chain, but rather as a crediblecommitment mechanism not to increase effort. Insurance allows the operatorto better coordinate effort along the supply chain. This is what we refer toas the strategic role of business insurance.As we can see from Figure 3.4, the introduction of insurance opportu-nities increases the welfare for the operator (relative to a scenario whereinsurance is not available). Conversely, the supplier’s profits decrease withinsurance. This is because insurance decreases the free-riding opportunitiesfor the supplier. We can also observe that the overall supply chain profitsincrease.Special Case OC: Operator with Wealth ConstraintsWe next move to the case where the operator has finite wealth, but thesupplier has no wealth constraints, i.e. WO < ∞ and WS = ∞. Therefore,constraint WCS never binds. In Proposition 7 we present our main resultsfor case OC, where we show that (as in case SC) insurance may be purchased583.4. Analysisfor strategic reasons. The intuition behind this result, however, is slightlydifferent from the one presented in the previous case.In the proposition below, we again show the existence of two wealththresholds. This time the thresholds, W IO and WIIO , are for the operator.The operator’s wealth constraint binds if WO is less than W IO and, in thissituation, the operator optimally increases the penalty rate that the supplierbears. The operator purchases insurance only when WO < W IIO .Proposition 7. Assume that WO < ∞ and WS = ∞. DefineW IO ≡ xΦUC(2yUC (1− β) + β)+ x(1− yUC)− 2 − pi and W IIO ≡xΦ (n, 0) (2n (1− β) + β) +x (1− n)−2−pi, where n satisfies (n(1−β))β−1x(β(1−n))β=(1 + β(1−2n1−n))2. The optimal contracting parameters are given by v∗ =vOC, y∗ = yOC and w∗ = wOC, where• vOC =0 if WO ≥W IO,0 if WO ∈ [W IIO ,WIO)(11−β)(y22 −(1−y)(WO+2−xy)(1−β)−xxβ) 12if WO < W IIO−β + 2y(34 + β)• yOC =yUC if WO ≥W IOΦOC (2 (1− β) + β) + 1− 2+WOx if WO ∈ [WIIO ,WIO)ΦOC−1−β(1−vOC)ΦOC−2 if WO < WIIO• wOC =wUC if WO ≥W IOxΦOCyOC + e∗S(yOC , 0)if WO ∈ [W IIO ,WIO)xΦOCyOC + e∗S(yOC , vOC)if WO < W IIOwhere ΦOC ≡ Φ(yOC , vOC).We illustrate Proposition 7 in Figure 3.5. When the operator’s wealthconstraint binds, the operator is unable to bear large financial losses. Tosatisfy this constraint, the operator needs to transfer away a higher share ofthe failure costs (relative to the unconstrained case UC). The operator cantransfer this share to the supplier, by increasing the supplier’s penalty rateabove yUC or, alternatively, transfer this share away from the supply chainthrough insurance.Similar to case SC, when the supplier internalizes the extra share of theliability (due to a higher penalty rate) not only does the supplier have larger593.4. AnalysisFigure 3.5: Optimal contract parameters - case OC, for β = 0.4, x = 100,pi =45.Figure 3.6: Welfare parameters - case OCincentives to increase effort, but the operator also has a larger incentive tofree-ride on the supplier’s efforts. This implies that any increase in the effortsof the supplier is partially offset by a further decrease in the efforts of theoperator. Therefore, the operator faces a trade-off between distorting theefforts in the supply chain (by increasing the supplier’s penalty rate, and ex-acerbating the free-riding problem) and dampening the supplier’s incentives(by purchasing insurance). However, in case OC the operator is the partythat free-rides on the effort of the supplier. This is unlike case SC, wherethe supplier is the free-rider.But if the operator is the one free-riding, why would it want to mitigatethe free-riding problem? The intuition is as follows. Note that the operator603.4. Analysiscoordinates the supplier’s efforts through the penalty rate, y, and compen-sates these efforts through a fixed-fee transfer w. Therefore, by optimallychoosing y and w, the operator extracts all rents in the supply chain. Whenthe free-riding problem becomes severe enough, the effort distortion becomesvery large. Because of this inefficiency the operator is able to extract fewerrents from the supplier.Due to the argument above, the operator optimally trades off the distor-tion in the supply chain efforts (by transferring the liability to the supplier) orthe dampening of the supplier’s incentives (by purchasing insurance). Whenthe operator’s wealth constraint is not stringent, a distortion of the supplychain efforts is preferred to a dampening of the supplier’s incentives. For thisreason, the operator chooses to increase the penalty rate, instead of purchas-ing insurance. However, when WO < W IIO , a dampening in the supplier’sincentives is preferred to an effort distortion (and the operator purchasesinsurance coverage). In Figure 3.6 we can observe that when WO < W IIO ,the operator’s (and, hence, the supply chain’s) welfare is improved throughinsurance.The General Case: Supplier and Operator with WealthConstraintsWe now assume that both the supplier and the operator are subject to wealthconstraints, i.e. that WS ≤ ∞ and WO ≤ ∞. In Proposition 8 we show theexistence of four regions. In the first region, none of the wealth constraintsbind. This leads to the unconstrained case UC. In the second and thirdregions, one of the wealth constraints is binding but the other is not. Theselead to the cases SC and OC.By looking at the general model, however, we must consider a new region.This is the region where both constraints bind at optimum. In this region theoperator purchases insurance, not for strategic reasons, but rather to ensurethe financial viability of the supply chain. In other words the operator’sproblem is not feasible without insurance. This is unlike the other regions,where the supply chain can run operations without insurance coverage.Proposition 8. Assume that WS ≤ ∞ and WO ≤ ∞ and define W˜ (y, v)≡x(1 + Φ (y, v) (β (1− y − v) + y (1− β)))−pi. The optimal contracting pa-rameters are613.4. AnalysisFigure 3.7: Contracting regions for the general case.• y∗ =yUC if WS > W IS and WO > WIOySC if WS≤W ISand WO > W˜(ySC , vSC)−WSyOC if WO ≤W IOand WS > W˜(yOC , vOC)−WOWSx(1−Φ∗) otherwise• v∗ =vUC if WS > W IS and WO > WIOvSC if WS≤W ISand WO > W˜(ySC , vSC)−WSvOC if WO ≤W IOand WS > W˜(yOC , vOC)−WO1− y (1 + β) otherwise−WS+WO+pi−xxβΦ∗• w∗ =wUC if WS > W IS and WO > WIOwSC if WS≤W ISand WO > W˜(ySC , vSC)−WSwOC if WO ≤W IOand WS > W˜(yOC , vOC)−WOxΦ∗y∗ + e∗S (y∗, v∗) otherwiseWe illustrate these results in Figure 3.7. In region 1 the wealth constraintsare non-binding. In region 2, WCO does not bind, but WCS binds at opti-mum. Note that in sub-region 2-I, we have thatWS ∈ [W IIS ,WIS). Therefore,by Proposition 6, the operator does not purchase insurance. Conversely,623.5. Extensionsin sub-region 2-II we have that WS < W IIS and the operator purchasesinsurance. In region 3, the supplier’s wealth constraint is non-binding,but the operator’s constraint is binding. In sub-region 3-I we have thatWO ∈ [W IIO ,WIO) and, by Proposition 7, the operator does not purchase in-surance. Conversely, in sub-region 3-II we have that WO < W IIO and the op-erator purchases insurance. In region 4, insurance serves a liquidity-enablingrole.3.4.3 Summary of ResultsIn §3.4.1 we show that a centralized supply chain only purchases insuranceas a way to ensure its financial viability (i.e. the liquidity-enabling role). Inother words, the centralized chain purchases insurance coverage if and onlyif the potential costs of an operational failure exceed the wealth of the chain,W = WS +WO.We show that the liquidity-enabling role also arises in a decentralizedsupply chain (see region 4 in Figure 3.7). However, in a decentralized settinginsurance is purchased even in situations where the supply chain has enoughwealth to cover any ex post losses associated with an operational failure (seeregions 2-II and 3-II). In these cases, the traditional explanations for whyfirms purchase insurance are absent, but nonetheless insurance is optimallypurchased. Our essay shows that in these cases insurance serves as a commit-ment mechanism to mitigate a free-riding problem. The free-riding problemarises when one of the firms has wealth constraints, but the other firm hassufficient wealth to sustain large losses.3.5 Extensions3.5.1 Sequential EffortIn the base model we assume that the efforts of the supplier and operatorare unobservable and, as we explain in Section 3.3.3, we can treat the effortsas simultaneous - even if these efforts are sequentially exerted. This is bythe Principle of Interchange of Moves (Osborne and Rubinstein, 1994).But the assumption of unobservability does not always hold. As a casein point, consider a supplier that is in charge of designing an equipment thatis subsequently managed by the operator: the supplier will be the first partyto exert effort, i.e. in the design of the equipment, and the operator willfollow, i.e. by exerting effort when operating the equipment. The equipmentoperator may be able to infer the effort of the supplier by inspecting the633.5. Extensionsquality of the equipment.30 In these situations it is reasonable to assume aStackelberg relationship in the exertion of the efforts.We now study an extension where the supplier leads in the exertion ofeffort, and the operator follows after observing the effort of the supplier (i.e.a Stackelberg dynamic). We begin by presenting the incentive compatiblelevels of effort in Lemma 9.Lemma 9. If the efforts of the parties are Stackelberg (and the supplierleads), the operator and the supplier exert effort levels e∗0,seq and e∗S,seq, wheree∗O,seq (y, v) =√√√√√x (β (1− y − v))2−β((1−β1+β)y)1−β − 1e∗S,seq (y, v) =√√√√√x((1−β)1+β y)1+β(β (1− y − v))β− 1From the lemma above, we can verify that the ratio of efforts is equal toe∗O,seq + 1e∗S,seq + 1=(1 + β)β(1− β)((1− y − v)y)(3.3)Note that the ratio of efforts obtained in the base model (from Lemma 3) isequal toe∗O+1e∗S+1= β(1−β)(1−y−v)y . By comparing these two ratios, we can noticethat when the supplier leads in the exertion of effort, there is an additionaldistortion in ratio of these efforts (i.e.e∗O,seq+1e∗S,seq+1= (1 + β)(e∗O+1e∗S+1)). Thismeans that for given levels of y and v, the Stackelberg advantage allows thesupplier to free-ride on the operator to a larger degree. We refer to this as aStackelberg distortion.So how does this assumption affect the results of base model? We findnumerically that if the supplier is the wealth-constrained party (i.e. case SC),the levels of insurance are greater in a Stackelberg relationship. Conversely,if the operator is the wealth constrained party (i.e. case OC), the levels ofinsurance are higher when the efforts are simultaneously exerted.This result is intuitive. To understand why, recall from the base modelthat in case SC, the wealth constraints of the supplier cause a distortion30Note that we are assuming that effort is observable but unverifiable in a court of law.This means that effort is not contractible. This is a commonly observed assumption (seepage 38 in Tirole, 1988). If the efforts of the firms are verifiable, these efforts can becontracted and the model becomes trivial.643.5. Extensionsthat increases the ratio of efforts between the operator and the supplier.So when the relationship is Stackelberg, this distortion is multiplied by theStackelberg distortion. Hence, the overall distortion is larger, and moreinsurance is needed to correct this problem.Conversely, recall that in case OC the wealth constraints of the operatorcause a decrease in the ratio of effort between the supplier and the opera-tor. But when effort is sequential, the Stackelberg distortion moves in theopposite direction, and counteracts the distortion generated by the wealthconstraints. As such, less insurance is needed to correct this problem.In all scenarios the supply chain profits decrease when the effort is sequen-tially exerted, given that the supplier (who is the agent) gains an advantageon the operator.3.5.2 Insurance Opportunities for the SupplierIn the base model we do not consider a scenario where the supplier (who isthe agent) is able to purchase insurance. This assumption allows us to focuson the coordinating role that insurance plays for business operators. Butthis assumption may not often hold in practice, as buying insurance is anoption that is also available to other parties in the supply chain.To address this issue, we study a setting where both the supplier and theoperator have the option of (independently) obtaining insurance coveragefrom third-parties.31 In this extension we seek to address two questions: (i)when, and for what reasons, would the supplier buy insurance? and; (ii) howdoes this possibility affect the strategic role of insurance for the operator?The following proposition allows us to answer both questions.Proposition 10. If the supplier has the option of purchasing insurance cov-erage:1. The supplier will choose coverage level v∗S =βy1+β .2. Under coverage level v∗S , the ratio of efforts between the operator andsupplier will be identical to the ratio given by equation (3.3) fromsection that in cases where (i) multiple parties are involved in an operational failure, and(ii) the degree of fault is ambiguous, insurance companies have ‘knock-for-knock’ agree-ments, where the insurers agree to reimburse the losses for which their respective policyholders are responsible in their procurement agreements, regardless of fault (Lilleholt et al.,2012).653.5. ExtensionsThe above proposition tells us that the supplier will purchase insurance re-gardless of whether the parties are wealth-constrained or not. Specifically,the supplier will use insurance as a commitment not to exert effort, whichwill drive the operator to increase its own effort. However, the use of insur-ance by the supplier will not be used to mitigate distortions in the effortsof the supply chain but, rather, to further distort these efforts in its favour.This is because the supplier is only concerned about its own profits, notabout the efficiency of the supply chain. The supplier thus uses insurance asa device to increase its capability to free-ride at the expense of the operator.Part 2 of the above proposition tells us that when the supplier buysinsurance, the ratio of efforts is identical to the case where the efforts aresequential. In other words, the supplier gains a Stackelberg advantage onthe operator by purchasing insurance. This is because insurance allows thesupplier to commit to a lower level of effort. This means that the analysisof this extension is qualitatively similar to the one from §3.5.1. Specifically,we find that the distortion generated by the supplier’s insurance coverageincreases the need for insurance for the operator in case SC, but decreasesit in case OC. The reasoning is similar to the one presented in §3.5.1.The supply chain profits decrease in all cases when the supplier has accessto insurance coverage. This is because the supplier (i.e. the agent) gains astrategic advantage on the operator and worsens the free-riding problem inthe supply chain. For this reason, it may be reasonable for an operator totry to impose a condition that supplier not purchase insurance, which isconsistent with the assumption made in the base model.3.5.3 Stochastic Failure CostsIn this section we relax the assumption that the failure costs are ex anteknown. To this end, let X ∈ {0, [xl, xh]} represent the failure costs. IfX = 0, operations have run as planned and the operator does not incur anycosts. If X > 0 a failure has occurred, and the costs are equal to x ∈ [xl, xh],where 0 < xl < xh. Assume that X has a probability atom at 0. Specifically,Pr (X = 0) = 1−F (eS , eO) where F (eS , eO) is defined as in the base model.By looking at the case where the failure costs are stochastic, we canexplore other settings. For example, tiered contracts (i.e. contracts wherethe penalty levels depend on the realized costs) are common in practice butnot considered in our main model. Second, under this assumption we canstudy the role of insurance deductibles in the insurance contract.663.5. ExtensionsTiered Contracts:Jain et al. (2013b) show that in the presence of financial constraints, con-tracts with tiered penalties are significantly more powerful in mitigatingdouble-moral hazard. This is because the design of tiered penalties allows theoperator to coordinate efforts without causing an excessive financial burdenon the wealth-constrained firm. For example, if the supplier has wealth con-straints, the operator can optimally increase the supplier’s penalty for smallfailures, and decrease the penalty for large failures. The expected penaltyfor the supplier thus remains unchanged and, at the same time, the finan-cial distress of the supplier is mitigated. For this reason tiered contracts arefrequently used in practice.32We assume that the operator designs two tiers, y1 ∈ [0, 1] and y2 ∈ [0, 1],and a tier threshold xT ∈ [xl, xh]. When X ≤ xT , the operator penalizesthe supplier with a penalty rate equal to y1X. When X > xT , the size ofthe penalty is y2X. The cash flows of the procurement contract (from theoperator to the supplier) areT (w, y1, y2, xT , X) ={w −Xy1 if X ≤ xTw −Xy2 if X > xTWe study this model by deriving some analytical results and by runningseveral numerical simulations. We first do comparative statics by varyingthe wealth of the parties, under the assumption that the failure costs areuniformly distributed. A set of results is illustrated in Tables 3.1 and 3.2. Inthese tables we present the optimal parameters under a contractual structureinvolving tiered contracts and insurance (xT , y1, y2; v). For comparison pur-poses, we also present the optimal parameters under a structure involving asimple penalty rate and insurance (y; v). In the right-most column of thesetables we present ∆Π, which shows by how much does the welfare of thesupply chain increase when the operator uses tiered contracts. We obtainthe following results.First, we find the operator can efficiently coordinate efforts through tieredpenalties and, at the same time, mitigate the impact of the wealth constraints32For example, in a joint operations agreement signed in 2001 between The Union Oilof California, the operator, and Ivanhoe Energy, the service supplier, the contractor usestiered penalties for any costs incurred in the event of oil spills, blowouts, fires, etc. Theoperator will be compensated for:(A) 5 % of total costs through $100,000; plus(B) 3 % of total costs in excess of $100,000 but less than $1,000,000; plus(C) 2 % of total costs in excess of $1,000,000. (Section III, clause 2)673.5. ExtensionsWO xT y1 y2 v y v ∆Π110 0 0.5 0.5 0 0.51 0 090 82.34 0.05 0.59 0 0.60 0 0.3270 100 0.41 0.70 0 0.71 0 1.0960 100 0.53 0.76 0 0.69 0.06 1.4950 100 0.60 0.75 0.08 0.67 0.16 2.0640 100 0.61 0.68 0.23 0.63 0.29 2.57Table 3.1: Optimal tiered and simple contracts - Case OCWS xT y1 y2 v y v ∆Π80 0 0.5 0.5 0 0.5 0 070 99.71 0.80 0.39 0 0.38 0 0.3860 100 0.57 0.28 0 0.27 0 1.0450 100 0.46 0.23 0 0.23 0.05 1.3940 100 0.17 0.17 0.06 0.17 0.13 1.5330 100 0.23 0.11 0.17 0.11 0.21 1.85Table 3.2: Optimal tiered and simple contracts - Case SC(i.e. the free-riding problem). However, when the wealth of the operator, orthe supplier, is small enough, a tiered contract is unable to eliminate exces-sive free-riding among the contractual parties. In these cases the operatorstill purchases insurance coverage, but the amount of coverage is smaller.Second, we study the sensitivity of our results to the variance of thefailure costs.33 We find that the role of insurance decreases when the varianceof the failure costs increases (see Tables 3.3 and 3.4). This is because whenthe failure costs have a large variance, the operator has more flexibility todistribute the failure costs through tiers. As such, tiered penalties can beused more effectively when the failures costs have a large variance.Insurance Deductibles:In this section we study the robustness of our model to more complex in-surance contracts, by allowing the operator to not only choose the level ofinsurance coverage, v, but also a deductible level, d ≥ 0. The insurance cash33To this end, we symmetrically shift xl and xh in opposite directions, so that xl + xhremains unchanged. For example, we look at a scenario where the expected failure costsare equal to 100, but we change the variance of the failure costs. To do this, we look atvarious cases: xl = xh = 100; xl = 75 and xh = 125; xl = 50 and xh = 150 and; xl = 25and xh = 175.683.5. Extensionsxl xh xT y1 y2 v25 175 93.73 0.59 0.73 050 150 95.48 0.63 0.75 0.0575 125 98.63 0.68 0.75 0.10100 100 100 0.76 0.76 0.14Table 3.3: Simulation of optimaltiered contracts for different dam-age variances (case SC)xl xh xT y1 y2 v25 175 93.72 0.25 0.13 0.0750 150 95.54 0.25 0.16 0.1175 125 98.13 0.25 0.20 0.14100 100 100 0.25 0.25 0.17Table 3.4: Simulation of optimaltiered contracts for different dam-age variances (case OC)flows are thus equal to I (v, d, P,X) = P −max {vX − d, 0}. If we assumethat the conditional probability of X, given X > 0, is uniform, then thecash flows of the insurance contract are given by EX [I (v, d, P,X) |eS , eO] =P − F (eS , eO)´ xhxlmax{vx−d,0}xh−xldxWe analyzed this model both analytically and numerically. We foundthat insurance deductibles are useless when the supplier has binding wealthconstraints, but the operator’s wealth constraint is non-binding (i.e. caseSC). In this case, there always exists an optimal insurance contract wherethe deductible is equal to 0. When the wealth constraint of the operator isbinding, and the wealth constraint of the supplier is non-binding (i.e. spe-cial case OC), positive deductibles allow the operator to mitigate free-ridingproblem more efficiently. Specifically, we find that insurance deductibles in-crease the range where insurance is strategically purchased. This is because apositive deductible allows the operator to seek high levels of insurance whenthe losses exceed its financial wealth. However, when these losses are smallenough, the (risk-neutral) operator does not benefit from insurance. There-fore, the operator uses a deductible to mitigate the impact of moral hazard,and seek high levels of insurance when needed (see Table 3.5). When the op-erator is wealth-unconstrained, and the supplier is wealth-constrained, thisrole is absent. The operator only benefits from deductibles when its wealthconstraints are binding.3.5.4 Alternatives to the Wealth ConstraintsThe wealth constraints used in the base model assume that the parties willnot enter the contractual relationship unless they are guaranteed to endup with non-negative wealth under all contingencies. These constraintsare common in practice and have been extensively analyzed in the litera-ture. In many cases, however, firms cannot entirely avoid the possibility of693.5. ExtensionsWO y v d y v ∆Π120 0.51 0 0 0.51 0 0100 0.60 0 0 0.60 0 0.3280 0.53 0.46 59.54 0.71 0 0.7070 0.54 0.46 48.71 0.69 0.06 1.4960 0.55 0.45 36.80 0.67 0.16 2.0650 0.55 0.44 23.24 0.63 0.29 2.57Table 3.5: Simulation of optimal insurance contracts with deductibles.bankruptcy but this does not prevent them from engaging in operationalactivities.Rather than avoiding the possibility of bankruptcy entirely, firms some-times adopt other criteria to maximize their profits and, at the same time,minimize their exposure to insolvency risks. To model this behavior, theliterature has considered a number of alternative approaches. In this sub-section we consider two popular approaches: the financial distress approachand the cost of bankruptcy approach.Financial Distress Approach: According to the financial distress ap-proach, firms may be averse to a state of insolvency, but are willing to toler-ate this possibility if the likelihood is sufficiently small. In other words, thefirm will engage in a contractual relationship only if the probability of insol-vency is below a tolerance threshold α ∈ [0, 1]. We write these constraints asfollows:Pr [ΠO (eO, w, y, v, P,X) +WO ≤ 0] ≤ αOPr [ΠS (eS , w, y,X) +WS ≤ 0] ≤ αSWe can easily verify that when the levels of tolerance (αO and αS) areequal to zero, the financial distress constraints are equivalent to the wealthconstraints. Through numerical results we show that when the tolerance todistress increases (i.e. when αS and αO increase), the levels of insurance aresmaller (see, Figure 3.8 for an illustration). In other words, the wealth con-straints can be seen as an extreme version of the financial distress approach.Overall, there are no qualitative changes in the results.Cost of Bankruptcy: Both the wealth constraints and the financialdistress constraints reflect the idea that firms only consider decisions wherethe risk of insolvency is either small or absent. Other papers consider thispossibility by incorporating the costs of negative wealth into the objective703.5. ExtensionsFigure 3.8: Optimal contract parameters for varying levels of α.function, rather than expressing them as constraints (See Greenwald andStiglitz (1990), Swinney et al. (2011) and references therein). Under thisapproach, we assume that the supplier and the operator maximize the utilityfunctions,US (eS , y, w,DS , X) ≡ ΠS (eS , w, y,X)−DSψS (eS , y, w,WS , X)UO (eO, w, y, v, P,DO, X) ≡ ΠO (eO, w, y, v, P,X)−DOψO (eO, w, y, v, P,WO, X)where DS and DO are the (exogenously given) bankruptcy costs foreach firm, and ψS (eO, w, y,WS , X) ≡ Pr {ΠS (eS , w, y,X) +WS ≤ 0} andψO (eO, w, y, v, P,WO, X) ≡ Pr {ΠO (eO, w, y, v, P,X) +WO ≤ 0} are thebankruptcy probabilities. UO and US are known as integrated objective func-tions.34 To analyze this setting, we consider a model where the operatorseeks to findmaxw≥0,y∈[0,1],v∈[0,1]EX [UO (e∗O, w, y, v, P (v|e∗S , e∗O) , DO, X) |e∗S ]34This approach captures the idea that reaching a state of insolvency brings out non-trivial costs. For example, the firms may be forced to sell their illiquid assets at low pricesto repay their debts. An insolvent firm will also have to pay for auditor and litigation feesin the event of filing bankruptcy.713.6. Conclusionsubject toEX [US (e∗S , y, w,DS , X) |e∗O] ≥ 0e∗S = arg maxeS≥0EX [US (eS , y, w,DS , X) |e∗O]e∗O = arg maxeO≥0EX [UO (eO, w, y, v, P,DO, X) |e∗S ]We performed numerical simulations using this model and found thatwhen the cost of bankruptcy is large for one party, but small for the other,the operator optimally buys insurance for its strategic value. Consider thecase where DO = 0. If DS is large, the supplier will weigh in the costs ofbankruptcy at the time of exerting effort. To mitigate this inefficiency, theoperator optimally decreases the penalty rate, which decreases the probabil-ity of bankruptcy for the supplier. In turn, this will generate a distortion inthe effort of both parties. When DS is too large, the operator is forced topurchase insurance as a way to commit not to increase effort and, thereby,to decrease the distortion in the effort among the parties. The intuition issimilar to the base model.Insurance will be purchased if the cost of bankruptcy is high for theoperator and low for the supplier, and vice versa. However, if the costs ofbankruptcy are high for both parties, insurance will not be purchased. Thisis because the distortion in the efforts of the operator is counteracted bythe distortion in the efforts of the supplier. In this case, the supply chainefficiency decreases, but the efforts are not distorted (i.e. there is no excessivefree-riding problem).3.6 ConclusionIn this essay we study a context where the interdependent (and unobservable)efforts of firms in a supply chain can mitigate the likelihood of an operationalfailure. We show that firms may purchase insurance for strategic reasons.This happens when one of the firms has severe wealth constraints, but theother firm has sufficiently large wealth to cover potential losses. In this sit-uation, contractual incentives alone leave the wealth-constrained firms withinefficiently low incentives to exert effort, and the “wealthy” firms with exces-sively high incentives. Because effort is substitutable, the wealth-constrainedfirm, which is aware of this incentive distortion, excessively free-rides on theefforts of the wealthy firm. Insurance coverage can mitigate this problemby transferring the failure costs (i.e. the financial liability) away from thesupply chain. Specifically, insurance allows the “wealthy” firm to credibly723.6. Conclusioncommit not to increase effort and this, in turn, decreases the incentives ofthe wealth-constrained firm to free-ride.The stylized model presented in this essay ignores some important oper-ational features. For example, in our model the firms are able to decreasethe likelihood of a failure, but not the magnitude of the failure costs (inthe event of a failure). In the literature, the first type of effort is knownas preventive effort, while the latter is known as contingency effort. If werelax this assumption, our main results would be affected if the contingencyefforts are effective enough to decrease the failure costs to a point where thewealth constraints become non-binding. However, this is highly unlikely inmany contexts, e.g. a nuclear meltdown, an oil spill or a product recall. Inthese contexts, an operational failure often causes financial distress or evenbankruptcy. The effectiveness of contingency efforts may, therefore, be verylimited.In this essay we demonstrate that business insurance may allow the sup-ply chain to operate more efficiently. Our results are particularly relevantto situations where large firms contract with considerably small suppliers.In these scenarios, a contractor would purchase business insurance, even ifthe contractor is wealth-unconstrained and risk-neutral. Through insurance,the contractors can prevent the wealth-constrained suppliers from excessivelyfree-riding on its reliability efforts. A similar situation arises when small con-tractors hire large suppliers. This implies that the availability of insurancehas non-trivial implications for supply chain contracting. To our knowledge,this role of insurance has not been previously highlighted in the literature.These results contribute to bridging the supply chain contracting and riskmanagement literatures.73Chapter 4Policy Incentives to Mitigatethe Impact of Operational TortLiability4.1 IntroductionIn industries characterized by the potential for injury or harm to third partiesdue to operational accidents, the costs of tort liability can be significant.35As a result of these costs, some firms may leave the market, and othersmay be discouraged from entering. Consider the following examples. Inthe 1980’s, the Diphtheria, Pertussis and Tetanus (DPT) vaccine, given tochildren, allegedly caused some cases of severe neurological damage. Aftera series of multi-million dollar lawsuits against the DTP vaccine producers,the insurance premium for vaccine liability rose dramatically. As a result,all but one of the vaccine manufacturers exited the market, and the vaccineprice went up by 6000 percent (Manning, 1994; Danzon and Sousa Pereira,2011). Similarly, over the last decade, major nuclear energy suppliers, includ-ing Westinghouse and General Electric, were reluctant to enter the Indianenergy market due to the costs associated with a potential nuclear accident(Bloomberg 2011). More recently, in 2013, a freight train from the Montreal,Maine & Atlantic (MM&A) Railway suffered a derailment in Lac-Mégantic,Quebec, resulting in the death of 47 people due to the explosion of 74-freightcars containing crude oil. After the crash, the railway company was unableto bear the liability and clean-up costs, and filed for bankruptcy. This led toconcerns in several cities in Maine, where the MM&A was the sole railwaycompany (Portland Press Herald, 2013).In situations like these, government intervention has often been necessary35In this essay, we use Shavell’s (2009) definition of an accident as a “harmful outcomethat neither injurers nor victims wished to occur, although the injurer... might haveaffected the likelihood of the outcome.” We also use Shavell’s definition of tort liabilityas a “legal obligation of a party who causes harm to make a payment to the victim of theharm.”744.1. Introductionto encourage market entry or to deter exit of firms. But there is widespreaddisagreement as to how firms should be incentivized. First, governmentscan provide incentives through ex ante subsidies. These subsidies influencedecisions that firms take prior to commencing dangerous operations. For ex-ample, the U.S. Nuclear Decommissioning Trust provides funds to supportthe safe decommissioning of reactors, and the Nuclear Waste Program Actof 1982 subsidizes the disposal of radioactive waste. Ex post subsidies helpmitigate the financial damages caused by the accident, by offering funds toshare the costs of a clean-up, or by limiting the firm’s exposure to liability.For example, both the U.S. Price-Anderson Indemnity Act of 1957 and theIndia Nuclear Liability Act of 2010 were created to protect suppliers againstnuclear accident liability. These acts impose a maximum cap on any ac-cident. Similarly, following the vaccination crisis of the 1980’s, the VaccineInjury Compensation Program created a no-fault liability system, which pro-tects producers of pediatric vaccines from liability costs. In the oil and gasindustry, the Oil Spill Liability Trust manages a $1.6 billion fund to helpcompanies pay for the clean-up costs of oil spills.In much of the law and economics literature, ex ante incentives are con-sidered superior to ex post incentives because of the moral hazard concernsassociated with the latter. This logic follows the idea that the firm must in-ternalize the full impact of any potential accidents to take the efficient levelof care to prevent accidents. By this view, offering ex post incentives willeffectively subsidize dangerous operations leading to an increased probabilityof an accident. Because ex ante subsidies are an up-front payment to thefirm, these do not diminish the financial impact of an accident for the firmand therefore do not induce moral hazard. Hence, ex ante subsidies are seenas the best way to induce sufficient market entry.However, as noted in the examples given earlier, both ex ante and ex postsubsidies are observed in practice. In this essay we demonstrate that whenthere is information asymmetry about the firms’ ability to prevent accidents,the provision of ex post subsidies may be efficient. Some firms may have aninherent advantage in improving the reliability of their operations, and thiscapability is either unobservable to a social planner (or prohibitively costlyto observe). This issue is central in the safety and regulation literature. Forexample, Antle (1996) has an extensive discussion on how uncertainty aboutthe ability of food manufacturers (to control their safety of their products)has led to market failure and inefficient regulation. This is because, when asocial planner does not have perfect information about the firms’ ability toreduce the likelihood of an accident, ex ante subsidies to invest in reliabilitywill go to all firms regardless of their ability to curtail operational accidents.754.1. IntroductionThis is socially inefficient because the entry decision of high-ability firms,who receive these subsidies, may be unaffected by these incentives.This essay seeks to characterize the conditions under which it is sociallyoptimal to use ex ante subsidies or ex post liability protection (or both). Toanswer this question, we model a market for a homogeneous good. This goodcan be supplied by two risk-neutral firms. The market is characterized byasymmetric information: both hidden information and hidden action. A firmcan either be a high-ability or low -ability type and this information is private.A high-ability firm can exert reliability improvements more efficiently (or,alternatively, at a lower cost) than a low-ability firm. In addition, theseinvestments are unobservable to the social planner and, as a result, they aresubject to moral hazard.The social planner has to determine a policy to maximize social welfare.He does so by choosing the level of ex ante subsidies and ex post liabilityprotection. The firms, depending on their type, choose whether to operatein (or exit) the market. The firms that stay in the market receive ex antesubsidies and invest in reliability. After operations begin, the firm(s) earnprofits and consumers gain utility from the good. At this stage, an accidentmay occur, and if it does, the firm will be liable for its share of the damage;this share is determined by the terms of the ex post incentives that had beenspecified at the outset.We demonstrate that in some cases it will be optimal to induce marketentry only by high-ability firms, and in other cases it is efficient to induceentry by all firms, independent of their ability to prevent operational acci-dents. This depends on three factors: (a) the level of market competition;(b) the potential accident costs and; (c) the opportunity cost of public funds.Specifically, if the welfare gains of increased competition are low, the acci-dent damages are high, or the opportunity cost of public funds is high, thenit is socially efficient to induce only high-ability firms to stay in the industry.In the converse scenarios it is optimal to induce both high- and low-abilityfirms to stay in the industry.Our results show that when it is socially optimal to induce only high-ability firms to stay in the market, the optimal policy will offer ex antesubsidies but no ex post liability protection. This is because, by offeringno ex post liability protection and optimally choosing ex ante subsidies, thesocial planner can ensure that low-ability firms will not find it optimal toenter the market. If, on the other hand, it is socially optimal to induceboth high- and low-ability firms to stay in the market, the optimal policywill offer a combination of ex ante and ex post incentives. Some level ofex post liability protection is efficient to provide incentives for low-ability764.1. Introductionfirms to enter the market. This is because relying on ex ante subsidies alonewould mean that higher ex ante subsidies would have to be offered to induceentry by low-ability firms. However, this provides excessive subsidies to high-ability firms. The social planner may therefore find it optimal to offer somelevel of ex post subsidies instead of higher ex ante subsidies, as a way toprovide an incentive for lower ability firms to enter the market.One key conclusion of the above analysis is the following. In some cases,the value of having multiple firms in the market is relatively large, or the costsof an operational accident are relatively low. In these scenarios, the socialplanner is willing to trade reliability for the benefits of increased marketcompetition. Here, the social planner will induce low-ability firms to stay inthe market and, as a result, he will offer both ex ante and ex post subsidies.In the converse scenario (i.e., if the costs of an accident are relatively high,or the benefits of market competition are low), then the social planner willvalue reliability more than market entry. In this case, the social planner willonly be willing to induce market entry by firms with high-ability and, as aresult, he will only use ex ante subsidies.This essay contributes to a long-standing debate about government in-tervention in industries that engage in “dangerous” operations. Coglianese(2010) argues that “ex post liability, while useful, does not always by itselfprovide socially optimal level of risk control. As such, preventative risk reg-ulation will be needed, and the core questions remain about how stringentshould such regulation be, and what form such regulation take. Risk regula-tion research will continue to be needed to provide conceptual clarity to thenormative basis for risk standards”. In this essay, we explore how govern-ment regulation, social welfare and operational safety are intertwined. Thisessay is of interest for policy makers, as it demonstrates the optimal combi-nation of ex ante and ex post mechanisms when entry and exit decisions arerelevant. We also show that the optimal intervention depends on the level ofindustry competition, the extent of the accident damages, or the opportu-nity cost of public funds. For the operations management community, thisessay is a first step to understanding how policy tools affect, not only theincentives to exert operational safety but also the structure of the industry,i.e. the number and type of competitors that firms will face. More broadly,this essay bridges the literatures on operational risk management and tortlaw.774.2. Literature Review4.2 Literature ReviewCalabresi (1970) is considered a seminal contribution to the economics andtort law literature. He describes three types of accident costs: primarycosts (direct accident losses), secondary costs (social costs of spreading - orconcentrating - the accident liability)36 and tertiary costs (judicial costs).In our essay, we analyze optimal policies in the presence of both primaryand secondary costs. Specifically, we study whether it is optimal to spreadaccident losses, through ex post liability protection, as a way to reduce thesocial costs of industry exit (even if this increases the expected primarycosts).Calabresi’s work is the preamble to a large debate on liability regula-tion. For example, Viscusi and Moore (1993) argue that liability protectionencourages beneficial innovation in R&D intensive industries and, at thesame time, Burk and Boczar (1993) show that liability protection is bene-ficial in the biotechnology industry, considering that manufacturers are ex-posed to other types of financial risks. However, Danzon and Sousa Pereira(2011) and Manning (1994) argue that liability protection has been eitherineffective, or even harmful, in the vaccine industry. Similarly, Trebilcockand Winter (1997) argue that liability protection has detrimental effects onsafety incentives and, unless regulation is a perfect substitute, it should notbe considered in the nuclear industry.In this essay, we focus on a context that combines asymmetric infor-mation and market exit. We show that the joint use of ex post liabilityprotection and ex ante subsidies may be optimal, even in the presence ofmoral hazard. In this sense, our paper is related to Kolstad et al. (1990)and Schmitz (2000), who look at the interaction between ex post liabilityand ex ante regulation. Like these papers, we show that ex ante and expost instruments are not always perfect substitutes. However, unlike thesepapers, we develop our model in a multiple-firm setting and, as a result, wefocus on industry wide equilibria and social welfare. Moreover, the papersabove focus on primary and tertiary costs, while our paper focuses on theinteraction between primary and secondary accident costs.36Secondary costs refer to the externalities that are generated when the accident liabilityis assigned to a specific party (or distributed across a group of parties). For example,suppose that courts rule in favour of making firms liable for all workplace accidents. As aresult of this ruling, firms may take two types of actions: (1) they may take precautionarymeasures (e.g. buy safety equipment their employees), or (2) they may decrease hiringrates for blue-collar jobs. Both actions will decrease primary accident costs. However,the second type of action may also increase unemployment in the economy, and should beaccounted for as a secondary cost.784.3. ModelOur paper also contributes to a small sub-field in the operations man-agement (OM) literature, one that looks at the interaction between policyincentives and operations. For example, Bakshi and Gans (2010) modelthe provision of government incentives in homeland security, by studyingthe Customs-Trade Partnership Against Terrorism (C-TPAT). Arora et al.(2008) analyze public policy in e-operations, in the face of potential vulnera-bilities to cyber security. Plambeck and Wang (2009) investigate the impactof e-waste regulation on new product introduction, and analyze the opti-mality “fee-upon-sale” and “fee-upon-disposal” policies. Kraft et al. (2013)look at both NGOs and government intervention, in industries where firmsmanage hazardous materials. Anand and Giraud-Carrier (2013) analyze andcompare three popular mechanisms to regulate pollution, by looking at theirimpact on social welfare. Cheung and Zhuang (2012) look at a similar con-text to the one analyzed by our model. In their paper, the authors lookat the impact of competition on optimal regulatory policies, in industriesmanaging dangerous operations. The authors find that the optimal level ofregulation is stricter under high degrees of competition. Our focus, in thisessay, is different in two ways. First, we focus on the interaction between exante and ex post incentives, while Cheung and Zhang focus on governmentenforcement (i.e governments ensuring that firms comply with regulatorystandards). Second, unlike this essay, the authors analyze optimal policiesunder given (i.e. exogenous) market structures. We focus on the way policyincentives affect not only operational decisions, i.e. investments in reliability,but also the structure of the market.4.3 ModelThere is a market for a homogeneous good, which can be supplied by two risk-neutral firms (i = 1, 2). The firms face a decision on whether to operate in themarket or to stay out. If a firm decides to operate, it faces the possibility thatit may cause an operational accident (e.g. a nuclear disaster). If an accidentoccurs, the firm is liable for the damages caused. To prevent this occurrence,the firm can invest in reliability. For example, it can conduct preventivemaintenance, purchase safety equipment or sample production batches forquality control. In reality, some firms are “better” at exerting reliabilityimprovements. For example, established firms may have a learning-by-doingadvantage over newer entrants. We assume that the firms fall into one oftwo categories: those with high-ability and those with low-ability at makingreliability improvements.794.3. ModelFigure 4.1: Model timelineFirms make reliability investments up to a point where the marginalexpected costs of an accident and the marginal investment costs are equal. Insome cases, however, the ex-post costs of an accident are high which would,in turn, require high reliability investments. These costs may discouragesome firms (especially those with low-ability) from operating in the market.If a firm chooses not to operate, social welfare decreases because there isless competition in the market. The social planner can correct this marketfailure by offering incentives, which encourage firms to stay in the industry.We consider two types of incentives: ex ante subsidies (to defray the costsof investing in reliability) and ex post liability protection (to defray accidentcosts). The social planner’s problem is to determine the policy that maxi-mizes social welfare, where social welfare is equal to producer surplus plusconsumer surplus minus the cost of public funds. Note that the govern-ment doesn’t know the exact ability type of the firms, but knows that theproportion of high-ability firms is p.Figure 4.1 summarizes the timeline of the model. This timeline can bedivided into four stages, which we explain in detail below.Stage 1: The Social Planner’s PolicyFirst, the social planner chooses the optimal policy, by offering either ex antesubsidies or ex post liability protection (or both). This policy is denoted bythe vector (s, b), where s ≥ 0 represents the level of ex ante subsidies andb ∈ [0, 1] represents the level of ex post liability protection, i.e. the proportion804.3. Modelof ex post costs paid by the social planner.37We let λ ∈ (0, 1) denote the marginal cost of public funds. In the publicpolicy literature, this term is also known as the shadow price of social funds(e.g., Laffont and Tirole, 1996; Dahlby, 2008). This term captures the factthat public funds allocated to this industry have alternative uses, i.e. there isan opportunity cost associated with providing incentives using public funds(Jones, 2005).Stage 2: Entry Decision and Market WelfareAfter observing the social planner’s policy, the firms determine their optimalentry strategy. The firms can operate, stay out or play a mixed strategy(i.e. operate with some probability). We let qi ∈ [0, 1] denote the entryprobability for firm i. We assume that there are no fixed costs for takingeither decision. For this reason, our results are invariant to whether the firmsare potential entrants or incumbents.The firms’ decisions will map into a market structure: duopoly (if bothfirms operate), monopoly (if only one firm operates) and complete mar-ket failure (if both firms stay out). Let X ∈ {D,M, 0} denote the marketstructure, where D represents a duopoly, M represents a monopoly and 0represents complete market failure. In other words, we have a mapping fromthe firms’ entry decisions to the market structure, (q1, q2)→X, such thatX (q1, q2) =D with probability q1q2M with probability q1 (1− q2) + q2 (1− q1)0 with probability (1− q1) (1− q2)(4.1)If only one of the firms operates in the market, it will receive monopolyrevenue ρM > 0 and if both operate, each will receive duopoly revenueρD∈[0, 12ρM]. In the monopoly case, consumer surplus is denoted by cM > 0and, in the duopoly case, consumer surplus is cD > cM . If both firms exit themarket, then the firms and the consumers will receive 0 profits and surplus,i.e. ρ0 = 0 and c0 = 0. As a result, market surplus will equal ρM + cM in amonopoly, 2ρD + cD in a duopoly and 0 if the market fails completely.The values for the above parameters are determined by a downward slop-ing demand function ∆ (P ). For example, if we consider the demand function∆(P ) = 1−P , we can verify that the monopoly outcome yields ρM = 14 andcM = 18 . Our model does not explicitly assume that (in a duopoly) the firms37We could, alternatively, model ex post liability protection by considering a liabilitycap. Both specifications are mathematically equivalent in our model.814.3. Modelcompete in a specific fashion, e.g. a Cournot or Bertrand competition. How-ever, to understand the implications of market competition on the optimalpolicy, we introduce a parameter κ ∈ [0, 1]. Here κ describes the degree ofcompetition among the duopolists. It ranges from perfect collusion (κ = 0)to Bertrand (perfect) competition (κ = 1). If κ = 0, then ρD = 12ρM andcD = cM and, if κ = 1, then ρD = 0 and cD > cM . Moreover, we assume thatfor any κ ∈ [0, 1], c′D (κ) > 0, ρ′D (κ) < 0. See figure 4.2 for an illustration.We assume any increase in competition increases market surplus, i.e. de-creases deadweight loss. Observe that 2ρ′D (κ)+c′D (κ) measures the increasein surplus (or the decrease in deadweight loss) as κ increases. Therefore, weassume that 2ρ′D (κ) + c′D (κ) ≥ 0 or equivalently,c′D(κ)2|ρ′D(κ)|> 1. Moreover,cD (1) captures the surplus of the good when there is no deadweight loss.To guarantee that our model is well-behaved, we assume that themonopoly surplus (cM + ρM ) yields at least half of the maximum marketsurplus which, in turn, guarantees us that 2cM + 2ρM ≥ cD + 2ρD, or equiv-alently, that 2cM − cD + 2 (ρM − ρD) ≥ 0. Note that this assumption is notvery restrictive, since it is satisfied by “well-behaved” demand functions (e.g.functions that are continuous, differentiable and downward sloping).38Stage 3: Reliability InvestmentsThe firms that enter the market will receive ex ante subsidies s, and makereliability investments. For example, the operator of a nuclear energy facilitycan improve the material quality of its containment areas, or increase thelevels of monitoring, as a way to ensure that there are no radioactive leakages.These efforts are only observed by the firm and denoted by r ∈ [0, 1].In addition to hidden action, we study a setting characterized by hiddeninformation. That is, we consider the case where some firms have greaterability to improve the reliability of their operations. For simplicity, we onlyconsider two types of firms, those with high-ability (H) and those with low-ability (L). When a firm has low-ability, the cost of effort is given by thefunction r22αHand if the firm has high-ability, the effort function is r22αL, forsome αH ≥ αL > 0. The parameters αH and αL measure the degree ofasymmetry between a high- and a low-ability firm.38For example, note that for a given demand function of the form D (P ) = (A−BP )m,given A,B > 0 and m ≥ 0, the profit maximizing price for a monopolist will be equalto P = AB(m+1) . Therefore, the ratio of monopoly surplus to total surplus will be equalto m−mm+1(m+ 1)m+1+ 1 > 12 .824.3. ModelFigure 4.2: An illustrative example using an inverse demand function,∆(Q) = 1−Q.Let τi denote the ability type of firm i. Each firm’s type is privateinformation, but the social planner and firm i have a prior about firm j’stype. Specifically, their prior that the probability that firm j has high abilityis equal to p ∈ [0, 1], and this prior is accurate.Stage 4: Operations and Potential AccidentsAfter reliability investments are made, operations will begin and accidentsmay occur. The probability that an accident occurs is equal to 1 − r. Thedamages caused are assumed to be equal to d ∈ (0,∞), where d encompassesa variety of costs that range from clean-up to litigation expenses. We assumethat d is deterministic. In section 4.6, we discuss the case where d is a randomvariable. In accordance with the policy offered by the planner, the firms willpay d (1− b) and the social planner will pay db.To avoid analyzing trivial scenarios, we assume that the accident damages(d) are high enough so that 2ρD + cD < 2d and ρM + cM < d. This allowsus to avoid cases where it is optimal to have firms in the market, regardless834.4. Analysisof their reliability investments. In other words, we assume that it is optimalnot to have a firm in the market, as opposed to entering the market andcausing an operational accident with certainty.4.4 AnalysisAnalyzing our model through backward induction, we first describe the op-timal level of reliability effort (in §4.4.1). Next, we determine the optimalentry decision (§4.4.2) and, finally, we characterize the socially optimal policy(in §4.4.3).4.4.1 Stage 3: Reliability Investment DecisionIn this stage, each firm will be either running operations, or be out of themarket. Let Πτii (ri|X, s, b) denote the stage 3 profits for firm i (whose abilitytype is equal to τi). We have thatΠτii (ri|X, s, b) = ρX︸︷︷︸Revenue− (1− b) d (1− ri)︸ ︷︷ ︸Expectedliability−r2i2ατi︸︷︷︸Costs of reliabilityinvestments+ s︸︷︷︸Ex antesubsidies(4.2)If the firm is out of the market, then Πi = 0. In stage 3, the firm’s problemis to findmaxri∈[0,1]Πτii (ri|X, s, b)Note that if firm i has ability τi, it will exert reliability investmentsrˆτii (b) ={(1− b) dατi if ατi ≤1d(1−b)1 if ατi ≥1d(1−b)Moreover, when the firms are not offered liability protection, i.e. whenb = 0, note that the firms fully internalize the accident costs and they willexert the level of reliability effort that minimizes primary accident costs.However, when ex post liability protection is offered, i.e when b > 0, thenthe firms decrease their reliability investments.844.4. AnalysisFigure 4.3: Illustration of the model regions.4.4.2 Stage 2: Optimal Entry DecisionIn stage 2, the firms play a simultaneous game where they make a decisionregarding q1 and q2. This decision is based on their expected profit level,which depends on the realized market structure. The stage 2 profits for thefirms are given byΠ˜τii (qi, qj |s, b) ≡ qi [qjΠτii (rˆτii (b) |D, s, b) + (1− qj) Πτii (rˆτii (b) |M, s, b)](4.3)Firm i chooses an entry strategy qi that maximizes its expected payoff,given the entry strategy qj of the rival firm. Observe that because the payoffto each firm is contingent on its own type τi, it is also contingent on the typeof the rival firm (τj), which is unknown to i. Recall, however, that firm i hasa prior p=Pr (τj = H).Let Bτii be the best response function for i. We have,Bτii(qHj , qLi |s, b, p)= arg maxqi∈[0,1]Π∗i (qτii , qHj , qLj |s, b, p) (4.4)where Π∗i (qτii , qHj , qLj |s, b, p) ≡ pΠ˜τii(qi, qHj |s, b)+ (1− p) Π˜τii(qi, qLj |s, b). Inthis game, a Bayesian Nash Equilibrium (BNE) is defined as a vector of entrystrategies that is a fixed point of the best response correspondences. Lemma11 presents this equilibrium.Lemma 11. Let firm 1 and firm 2 have ability type τ1 and τ2, respectively,and let mτ ≡ d (1− b) (1− rˆτ ) +rˆ2i2ατrepresent the expected firm liability854.4. Analysisplus reliability investments. The Bayesian Nash Equilibrium for the entrystrategies of firm 1 and 2 is(qBNE1 , qBNE2)= (qˆτ1 , qˆτ2) where,• qˆH =1 if s+ ρM −mH > p (ρM − ρD)s+ρM−mHp(ρM−ρD)if s+ ρM −mH ∈ [0, p (ρM − ρD)]0 if s+ ρM −mH < 0• qˆL =1 if s+pρD+(1−p)ρM−mL1−p > ρM − ρDs+pρD+(1−p)ρM−mL(1−p)(ρM−ρD)if s+pρD+(1−p)ρM−mL1−p ∈ [0, ρM − ρD]0 otherwiseLemma 11 presents three regions of parameter values; these regions areillustrated in Figure 4.3. In this example, we have that ρM = 104 and cM =108 , ρD =109 , cD =209 , d = 6,αH =1200 , αL =1210 and λ = 0.1. In the figureabove, region 0 represents a scenario in which both firms exit the market,i.e.qH = 0 and qL = 0. In regions H1 and H2, high-ability firms will choosea positive entry probability, while the low-ability firms will choose to exit.Specifically, in region H1 the high ability firms set qH = 1pρM+s−mHρM−ρDwhilelow-ability firms set qL = 0 and, in region H2, the high-ability firms set qH =1 and low-ability firms set qL = 0. Finally, regions HL1 and HL2 representthose scenarios where both high- and low-ability firms choose a positive valuefor qi. Specifically, region HL1 represents those scenarios in which high-ability firms set qH = 1 and low-ability firms set qL= 11−ps−mL+pρD+ρM (1−p)(ρM−ρD)and, region HL2 represents those scenarios where the high-ability firms setqH = 1 and the low-ability firms set qL = 1. Finally, note that the probabilitythat high- and low-ability firms enter the market is a decreasing function ofthe prior p.4.4.3 Stage 1: The Optimal PolicyIn stage 1, the social planner seeks to maximize social welfare (W), whichis defined as firm profits plus consumer surplus minus the opportunity costof public funds. To characterize social welfare, let us first look at stage 3.Suppose that, at the beginning of this stage, a monopoly structure has beenrealized (i.e. X = M) and that the monopolist (firm i) has ability type τi.In this case, the stage 3 social welfare will be equal toWτi (M, s, b) = Πτii (rˆτii (b) ,M, s, b)︸ ︷︷ ︸Firm profits+ cM︸︷︷︸Consumer welfare− (1 + λ) [db (1− rˆτii (b)) + s]︸ ︷︷ ︸Cost of public funds(4.5)864.4. AnalysisSimilarly, suppose that at the beginning of stage 3, a duopoly structurehas been realized, and that the duopolists have ability τ1 and τ2. Notingthat WH,L (D, s, b) =WL,H (D, s, b), we haveWτ1,τ2 (D, s, b) =∑2i=1 Πτii (rˆτii (b) , D, s, b) + cD- (1 + λ)[db∑2i=1 (1− rˆτii (b)) + 2s] (4.6)Finally, if the market fails completely, we have that W (0, s, b) = 0.As a result, we can write the expected social welfare (in stage 1) asE [W (s, b) |p] = 2(p(1− qˆH)+ (1− p)(1− qˆL)) (pqˆHWH (M, s, b))2(p(1− qˆH)+ (1− p)(1− qˆL))(1− p)qˆLWL (M, s, b)+p2(qˆH)2WH,H (D, s, b) + 2p (1− p) qˆH qˆLWH,L (D, s, b)+ (1− p)2(qˆL)2WL,L (D, s, b)The social planner’s problem is to findmax(s,b)E [W (s, b) |p]To characterize the optimal policy, we first demonstrate the existence ofthree different regions. Consider Lemma 12 below (which is illustrated inFigure 4.4).Lemma 12. Let (s∗, b∗) = arg max(s,b)E [W (s, b) |p]. There exists α1≥ 0and α2 ≥ 0 such that:1. (Region 0) If αH ≤ α1, then (s∗, b∗) will induce both high-ability andlow-ability firms to exit the market, i.e. to choose qi = 0.2. (Region H) If αH > α1 but αL ≤ α2, then (s∗, b∗) will induce low-ability firms to exit the market, i.e. to choose qi = 0. Conversely,(s∗, b∗) will induce high-ability firms to enter with positive probability,i.e. to choose qi > 0 .3. (Region HL) If αL > α2, then (s∗, b∗) will induce high-ability firms toenter, i.e. to choose qi > 0, and low-ability firms to enter with positiveprobability i.e. to choose qi > 0.Lemma 12 tells us that, whenever αH ≤ α1, then the optimal policy willlie in region 0 (of figure 4.3). Conversely, if αH > α1 but αL ≤ α2, then theoptimal policy will lie in either region H1 or H2. However, if αL ≥ α2, thenthe optimal policy will lie in either region HL1 or HL2. The intuition behind874.4. AnalysisFigure 4.4: A visualization of Lemma 12this result is straightforward. Suppose that αH and αL are very large. Inthis case, both firms are very inefficient at exerting reliability improvementsand, all else equal, they will choose low values for ri. If this happens, itis better to “shut down” the market than to incentivize unreliable firms tooperate. The second region in Lemma 12 presents a scenario where the highability firms are efficient, but the low-ability firms are inefficient. In thisscenario, it is optimal to incentivize only high-ability firms to operate in themarket. Finally, in region HL, the lemma presents a scenario in which bothtypes of firms are efficient. In this region, it is optimal to induce both firmsof high- and low-ability to enter the market with positive probability.Region 0 is trivial in the sense that the social planner will find it optimalto offer no incentives. For this reason, the rest of this essay focuses onthose scenarios where αH ≥ α1, i.e. on those scenarios where it is sociallyoptimal to provide incentives so that, at least, the high-ability firms enterthe market. In the lemma below, we characterize the optimal policy for theseregions. Without loss of generality we present results for the case where, atequilibrium, rˆi (s∗, b∗) is less than one. In other words, we focus on interiorsolutions.Proposition 13. Suppose αH ≥ α1. The optimal policy, (s∗, b∗), can becharacterized as follows:884.4. Analysis1. (Region H) If αL ≤ α2, then (s∗, b∗) =(sH, 0), wheresH = d−αHd22− ρM (4.7)+ (ρM − ρD)mincM + (1 + λ)(ρM − d+αHd22)2 (1 + λ) (ρM − ρD (κ)) + (2cM − cD (κ)), p(4.8)2. (Region HL) If αL ≥ α2, then (s∗, b∗) =(sHL, bHL), wheresHL =(1− bHL)d−αL((1− bHL)d)22− ρM (4.9)+(ρM − ρD)[cM + (1 + λ)(ρM − d+ αL(1− bHL)d2(1− 1−bHL2))]2 (1 + λ) (ρM − ρD (κ)) + (2cM − cD (κ))and bHL ∈ (0, 1) , satisfying the first order condition(1 +λ(1− 2(1− bHL))bHL)(p (αH − αL)αL (1 + λ))(4.10)= −cM1+λ +(ρM − d+ αL(1− bHL)d2(1− 1−bHL2))2 (ρM − ρD (κ)) +(2cM−cD(κ))1+λ(4.11)The intuition behind Proposition 13 is the following. In region H, the socialplanner wants to institute a policy so that only high-ability firms choose apositive entry probability. Through this policy, any entrant will reveal itstype (as a high-ability firm) and, for this reason, there will be no hiddeninformation about the entrant’s ability. Furthermore ex ante subsidies donot distort a firm’s incentives to invest in reliability, while ex post liabilityprotection does. As a result, ex ante subsidies are superior to ex post liabilityprotection.Let us now turn to region HL. In this region, it is optimal to provide in-centives so that both high-ability and low-ability firms choose to enter withpositive probability. If the policy were solely based on ex ante subsidies,it would provide informational rents to high-ability firms. That is, to en-courage entry among low-ability firms, the high-ability firms will necessarilyreceive incentives in excess of their needs. In other words, the social plan-ner could decrease the level of ex ante incentives, by some small quantity,without affecting the entry decision of the high-types, but affecting the en-try strategy of low-ability firms. Therefore, it is optimal to substitute somelevel of ex ante subsidies for ex post liability protection. Specifically, theoptimal policy will balance out the cost of informational rents (caused byex ante subsidies) with the cost of hidden action (caused by ex post liability894.5. Comparative Staticsprotection). In summary, the optimal policy balances out three components:the welfare gains from increased entry, the costs of hidden information, andthe costs of moral hazard.4.5 Comparative StaticsIn this section, we analyze the factors that determine whether the optimalpolicy lies in region H or HL. We consider three factors: (i) the degree ofmarket competition κ, (ii) the magnitude of the accident damages d, and(iii) the marginal cost of public funds λ.4.5.1 Industry CompetitionThe level of market competition is a key factor in the analysis of market entryand social welfare (see Cabral 2004). Specifically, the level of competitionmeasures the social value of having multiple firms in the industry. Whenfirms in the industry compete aggressively, the cost of firm exit is muchhigher to society than it is to the firm, and vice versa. This is becausewhen there is aggressive competition, a firm that enters the market willsignificantly benefit consumers, while only earning modest economic rents.To see how the level of competition affects the optimal policy, considerLemma 14 below. Note that when duopoly competition (κ) increases, thenthe consumer surplus increases by c′ (κ), and firm profits decrease by ρ′D (κ).Lemma 14. If c′D(κ)2|ρ′D(κ)|− 1 ≤ λ, then ∂qˆL(s∗,b∗)∂κ ≤ 0. Ifc′D(κ)2|ρ′D(κ)|− 1 > λ, then∂qˆL(s∗,b∗)∂κ ≥ 0.Recall that by our assumption that increased market competition alwaysincreases market surplus (i.e. decreases deadweight loss), we getc′D(κ)2|ρ′D(κ)|−1 ≥0. Lemma 14 tells us that when the marginal increase in market surplus (fromincreased competition) is less than the opportunity cost of public funds, thenthe social planner will be less inclined to induce firms of low ability into themarket, and vice versa. Lemma 14 leads us to the following proposition.Proposition 15. When the surplus gains of increased competition are small,the social planner will only offer ex ante subsidies (but no ex post liabilityprotection). Conversely, when the surplus gains of increased competition arelarge, the social planner will offer ex post liability protection.Proposition 15 yields the following intuition. Suppose that market com-petition is very aggressive. In this case, consumer surplus will be high in a904.5. Comparative StaticsFigure 4.5: Illustrative example with inverse demand function P (Q) = 1−Q.duopoly. Conversely firm surplus will be low. In other words, consumers willgreatly benefit from having two firms in the markets, but the firms will seetheir profits considerably reduced by the aggressiveness of competition. Asa result, they will be less motivated to enter the market. On the other hand,the social planner will be more inclined to incentivize the firms to enter themarket, to achieve a duopoly, and reduce the deadweight loss. As a result hewill offer a more “generous” policy. The proposition above tells us that whenthe gains from increased competition are very large, then the generosity ofthis policy will be high enough so that low-ability firms are inclined to enterthe market (i.e. he will offer a policy that lies in region HL). And, as wesaw in Lemma 13, when the policy is in region HL, the social planner willoffer ex post liability protection. If, conversely, the policy lies in region H(i.e. when the surplus gains from competition are low), the social plannerwill only offer ex ante subsidies.An alternative interpretation to the proposition above is the following.When the duopolists compete in such a way that it increases welfare signifi-cantly, then the social planner will be willing to trade reliability (by inducinglow-ability firms to operate) for the benefits of competition. As a result, thesocial planner will offer ex post liability protection. See Figure 4.5 for anillustration.4.5.2 Accident DamagesIn this subsection, we analyze the impact that the accident damages haveon the optimal policy. We begin with the following proposition.914.5. Comparative StaticsProposition 16. When the accident damages, d, are small, the optimalpolicy will offer a combination of ex ante subsidies and ex post liability pro-tection (i.e. (s∗, b∗) will lie in region HL). Conversely, when the accidentdamages are large, the optimal policy will only offer ex ante incentives (i.e.(s∗, b∗) will lie in region H).The intuition behind Proposition 16 is the following. When a firm decidesto enter the market, not only does it bring surplus, it also introduces a burdento society. In other words, the entry of a firm raises expected accident costs.When the social planner induces entry by low-ability firms, he is willingto trade reliability in exchange for market efficiency. However, when theaccident damages are large, then the social planner is less willing to makethis tradeoff. When this happens, the social planner will find it optimal toinduce entry only among high-ability firms, by offering ex ante subsidies andno ex post liability protection. See Figure 4.5 for an illustration; note thatthe y axis represents the accident damages d.4.5.3 Marginal Cost of Public FundsThe marginal cost of public funds λ refers to the opportunity cost of trans-ferring funds to the firms. In this subsection, we analyze the impact thatthis parameter has on the optimal policy. Consider Proposition 17 below.Proposition 17. Let λ ∈ [0, 1] be given. We have that:1. When the marginal cost of public funds λ is small, the optimal pol-icy will offer high levels of ex ante subsidies, but low levels of ex postliability protection.2. When λ is intermediate, the optimal policy will offer intermediate levelsof ex ante subsidies and high levels of ex post liability protection.3. When λ is large, then the optimal policy will offer low levels of ex antesubsidies, and no ex post liability protection.If the marginal cost of public funds is low, the social planner will offerlarge amounts of ex ante subsidies, but low amounts of ex post liabilityprotection. To understand why, let us consider an extreme scenario, whereλ = 0. Here, there is no welfare loss attributable to transferring funds tothe firms (i.e. a $1 loss from the public budget is offset by a $1 gain by thefirms). In this case, the rent extraction problem that arises with ex antesubsidies disappears. In other words, the social planner will be indifferent924.6. Extensions, Discussion and ConclusionFigure 4.6: Optimal policy as a function of λ.to the fact that high-ability firms extract rents from public funds. As λincreases to intermediate amounts, the rents extraction problem increases,and the social planner will find it optimal to decrease the level of ex antesubsidies, in exchange for some level of ex post liability protection.When λ is large, the opportunity cost of public funds becomes significant.Here, the social planner will find it very costly to provide incentives to thefirms and, as a result, it will become too costly to incentivize low-abilityfirms. When this happens, the optimal policy will shift to region H, and thesocial planner only offer ex ante subsidies (but no ex post liability protection).See figure 4.6 for an illustration.4.6 Extensions, Discussion and ConclusionThis essay bridges the operational risk management and the public policyliteratures by studying the impact of potential operational accidents on mar-ket entry, and the role of government intervention in correcting any resultingmarket failure. It also contributes to a large debate in the public policy lit-erature: are ex ante incentives always superior to ex post incentives, or viceversa? Our results show that when there is information asymmetry aboutthe firms’ ability to prevent accidents, the provision of ex post subsidies maybe efficient. This is because a social planner cares not only about invest-ments to make operations safer, but also about having the socially optimalnumber of firms enter the industry. Because firms only care about their934.6. Extensions, Discussion and Conclusionprivate profits, and not social welfare, forcing firms to fully internalize thecosts of accidents may lead to insufficient entry. By appropriately choosingex ante and ex post subsidies the social planner can balance the impact oftwo externalities: the externality created by an accident and the externalitycreated by insufficient entry.The above results, however, are obtained by analyzing a highly stylizedmodel. For example, our model does not consider multi-period interactions,demand-contingent or stochastic accident damages or the role of insurancemarkets. Below, we discuss the robustness of our model to the introductionof these elements.Stochastic Accident DamagesOur model studies a setting where the accident damages are deterministicand known ex ante. However, our results are also robust to scenarios wherethe accident damages are stochastic. To understand why, note that ourmodel focuses on high-impact and low-probability accidents, e.g. oil spillsand nuclear meltdowns. The probability distribution of these accidents istypically characterized by a very long tail, which includes devastating ac-cidents (i.e. black-swan events). In these scenarios, firms typically assesstheir decisions, not on the extent of the accidents, but rather on the Valueat Risk (VaR), i.e. on the probability that an accident realization will besufficiently high to jeopardize the firm. For example, suppose that a firmhas a $10 billion net worth. An oil spill that results in clean-up costs abovethis amount will jeopardize a firm’s existence, whether the net cost of thisspill is $15 or $20 billion. Under this argument, we could re-interpret ourmodel by assuming that d measures the value that would place the firm atfinancial risk, and not the accident damages.Demand-Contingent Accident DamagesOur model also assumes that the accident damages are independent of de-mand. For example, in the childhood vaccine industry the extent of the acci-dent damages is highly (or even perfectly) correlated to the market demand.In this case accident damages will depend on industry output, which in turndepends on whether the industry is monopolized or is a duopoly. To modelthis setting we should assume that the magnitude of the accident damagesis dependent on industry structure. Under this assumption, the monopolyoutcome becomes “less desirable” to the firms, as more demand also implieshigher potential accident costs. Conversely, the monopoly outcome may be-944.6. Extensions, Discussion and Conclusioncome more desirable to the social planner because less demand implies lessaccident costs. However, the fundamental tradeoff between market entryand operational reliability is still present, which leads us to conjecture thatthe insights will be unaltered by this extension.Menu of Policies In our model, we consider a context where the socialplanner’s intervention is characterized by a single policy. An alternativewould be to take a mechanism design approach, which would yield a menu ofpolicies. This approach, however, would have not affected the main insightsof our model. First, if the social planner wishes to incentivize entry by high-ability firms only, then it is optimal to have a single policy targeted at thehigh-ability firms (which would not be attractive to low-ability firms). If, onthe other hand, the social planner wishes to induce entry by firms with high-and low-ability, then he would have offered a menu containing two policies.The first policy, which would be incentive compatible with the high-abilityfirms, would offer ex ante subsidies, but not ex post incentives. This policywould provide positive rents to the high-ability firms. The second policy,which is incentive compatible with the low-ability firms, would offer somedegree of ex post incentives, and a lower level of ex ante subsidies. Thispolicy, however, would provide no rents for the low-ability firms. While thiswill be a more efficient intervention, implementing a menu of contracts isoften infeasible in public policy. The key point, however, is that the trade-off between rent extraction and efficiency would also exist in the mechanismdesign approach.Multi-Period DynamicsThe scope of our analysis is restricted to a single period setting. If we wereto consider a dynamic model with entry and exit decisions in each period,then the optimal policy would need to address issues such as bankruptcy,which are absent in a single period model. That is, if the firms have limitedwealth, the possibility of (costly) operational accidents does not only affecttheir entry decision, but also their ability to operate after the occurrence ofa potential accident. For example, if the cost of an operational accident (d)is stochastic, and large realizations of d are possible, then the firms will beforced to exit the market after the occurrence of a such a realization. In thissetting, the social planner will not only need to address issues related to exante entry (i.e. firms deciding whether to undertake operations), but also expost entry and exit (i.e. firms deciding whether to continue in the marketafter causing an operational accident).954.6. Extensions, Discussion and ConclusionThe introduction of a dynamic model also introduces the possibility oflearning-by-doing effects. In the presence of these effects, the social plannermay be more willing to tolerate the entrance of low-ability firms, especiallyduring the initial periods. This implies that the social planner may be ini-tially willing to offer higher levels of ex post liability protection (to induceentry by low-ability firms) and, as these firms converge towards a high abilitylevel, the social planner may progressively switch to a policy that is purelycharacterized by ex ante subsidies.Liability InsuranceWe did not explicitly highlight the role of insurance markets. In practice,there are two possible roles that an insurer would play. First, the insurercould provide coverage for any “black-swan” events that impose costs abovethe accident damages d. This feature can easily be accommodated in ourmodel, by simply adding an (ex ante) insurance premium to the firm’s profitfunction. Under this interpretation, d would represent the insurance de-ductible.A second role for insurance could be to provide liquidity to the firm tocover ex post accident costs. In our current model we assume that the firmself-insures, i.e. it has access to enough cash reserves to pay its share of expost costs in the event of an accident. If there exists a perfectly competitiveinsurance market, and the two parties are able to write a complete contract,then the firm has access to sufficient liquidity in the event of an accident.In this case, we can interpret the firm not as a single entity, but rather a“firm-insurer” pair. This interpretation will fail, however, if there are anyfrictions in the insurance market: for example, if the insurance market is notcompetitive or if transaction costs are non-trivial. In these circumstances,the social planner would need to consider the role of the insurance marketand will need to respond to any inefficiencies in this market.96Chapter 5ConclusionsThe first essay bridges the firm-level productivity and supply chain litera-tures. This essay provides new evidence about the link between Total FactorProductivity and supply chains. It does this by identifying the various mech-anisms through which productivity can spillover across firms. Our resultscan be useful for practitioners at the time of managing their supply chain re-lationships. To arrive at this result, however, our essay first had to deal withseveral identification issues, some which are considered extremely challeng-ing in the econometrics literature. To this end, we used a novel econometricapproach that allowed us to overcome these challenges. This methodologycould be adopted in future research, in order to identify the mechanismsthrough which different types of spillover effects propagate across verticalrelationships.The second essay (Chapter 3) expands our view on the roles of businessinsurance, by showing that business insurance may allow the supply chain tooperate more efficiently. These results are particularly relevant to settingswhere large firms contract with small suppliers, and vice versa. The unevendistribution of wealth would lead to firms being more likely to buy insuranceas a strategic tool. This role of insurance has not been previously highlightedin the literature. Our results thus imply that the availability of insurancehas non-trivial implications for supply chain contracting (at least when theserelationships are asymmetric). In general, our results imply that insurancecoverage and contractual incentives are not necessarily substitutes, but mayrather complement each other. These results contribute to bridging thesupply chain contracting and risk management literatures.The third essay (Chapter 4) contributes to a long-standing debate aboutgovernment intervention in industries that engage in “dangerous” operations.Coglianese (2010) argues that “ex post liability, while useful, does not alwaysby itself provide socially optimal level of risk control. As such, preventativerisk regulation will be needed, and the core questions remain about howstringent should such regulation be, and what form such regulation take.Risk regulation research will continue to be needed to provide conceptualclarity to the normative basis for risk standards.” Our paper is of interest for97Chapter 5. Conclusionspolicy makers, as it demonstrates the optimal combination of ex ante andex post mechanisms when market exit decisions are relevant. We also showthat the optimal intervention depends on the level of industry competitionand on the extent of the accident damages. 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J DavidCummins, Georges Dionne, eds., Handbook of Insurance. Springer Nether-lands, Dordrecht, 155–183.106Appendix ATechnical Results and VariableDefinitions of Chapter 2A.1 The Olley-Pakes ApproachWhen we estimate TFP through OLS, the resulting estimates are likely tosuffer from simultaneity and selection biases. A simultaneity bias arisesbecause there is systematic correlation between the input factors and theerror term. A selection bias arises because a firm’s profitability is correlatedto its level of capital stock, which is fixed in the short term.To deal with these problems, Olley and Pakes (1996) introduced a semi-parametric specification that controls for both biases. This approach usescapital investments, Iit, as a proxy variable, and makes the following as-sumptions (which are grounded on empirical results; see Olley and Pakes,1996) :1. Labour (lt) is a variable factor at time t.(a) Capital (kit) is a fixed factor at time t, and a function of theproductivity level at t− 1, i.e. kit = k (ρi,t−1).(b) Iit is a function of ρit and kit, i.e. Iit = I (ρit, kit). This functionsatisfies ∂Iit∂ρit> 0 for any Iit > 0.By assumption 3, we have that ρit = h (Iit, kit), where h is the inversionof I. Therefore, we can re-write equation (2.1) asyit = βllit + φit (Iit, kit) + εitwhere φit (Iit, kit) ≡ h (Iit, kit)+βkkit.39 Note that φ isolates ρ, which isthe source of the simultaneity bias. For this reason, we can estimate yitand obtain consistent estimates for βl. Although φ is unobservable, we39For simplicity of exposition, we ignore industry fixed effects. In the actual calculation,we include these terms.107A.2. The Formation of Supply Chain Networksestimate this function by using a third-order polynomial expansion onI and k, i.e. φit (Iit, kit) ≈ c0 +∑3m=0∑3n=0 cmnkmit Init. This estimationyields estimates βOPl and φˆit.Second, we estimate βk. To this end, let ∆ρit ≡ ρit−ρi,t−1 and assumethat cov(kit,∆ρit) = 0. Thus,yit − βOPl lit = βkkit + ρit + εit= βkkit + ρi,t−1 + ∆ρit + εit= βkkit + φˆi,t−1 − βkki,t−1 + ∆ρit + εitwhere φˆi,t−1 − βkki,t−1 is an unbiased estimate of ρi,t−1.Now, define ρit(Ii,t−1, kit) as the productivity threshold for which acompany is indifferent between exiting the market and running oper-ations. Also, assume that probability of survival of firm i, Pit, is afunction of ρi,t−1 and ρi,t−1. To estimate Pit we run a probit regres-sion on a third order polynomial expansion on Ii,t−1, ki,t and the firm’sage.We use the estimated survival probability, Pˆit, to estimateyit − βOPl lit = βkkit + g(φˆi,t−1 − βkki,t−1, Pˆit)+ ∆ρit + εit,where g (·) is an unknown function. To approximatethis function, we use a third order polynomial expan-sion on its parameters, where g(φˆi,t−1 − βkki,t−1, Pˆt)≈c0 +∑3m=0∑3n=0 cmn(φˆi,t−1 − βkki,t−1)mPˆnit .This allows us to obtain βOPk . We use βOPl and βOPk to derive thefirm’s productivity, where,ρit = yit − βOPk kit − βOPl litA.2 The Formation of Supply Chain NetworksIdentification StrategyTo control the selection of supply chain partners, we build an approach thatjointly estimates two types of processes: (1) the selection process and; (2) thedeterminants of productivity (i.e. the firm effects and the spillover effects),conditional on the selection process. We explain these two processes belowand, thereafter, our estimation technique.108A.2. The Formation of Supply Chain NetworksProcess 1: Selection of Supply Chain PartnersA link between a supplier and a customer is formed as a result of two choices,i.e. the customer selecting a potential supplier, and the supplier agreeing totrade with the customer. To simplify our model we assume that suppliersare always willing to trade with any customer. Although this assumptiondoes not always hold,40 it is often the case that customers are the dominantdecision makers at the time of forming links.Consider the network matrix C, and recall that this matrix has typicalelement cij = 1 if j is a customer of i, and cij = 0 otherwise. At thebeginning of period t, firm j evaluates the expected profitability of forminga link with supplier i, Uj (i). We assume that j selects i as a supplier if andonly if Uj (i) > 0. In other words,cij = 1Uj(i)>0The term Uj (i) depends primarily on the compatibility between the firms’industries. For example, manufacturers of home furniture are likely to requireinputs from firms in the wood and lumber industry, but less likely to sourcefrom, say, firms in the air travel industry. The term Uj (i) also depends onthe observed characteristics of both the customer and the supplier, Xi andXj , and on the state of their relationship in the previous period. Specifically,letUj (i) = α0 + αcc0ij + αV V0ij + αgeogeoij +∑kαkxki + ηij (A.1)The variable c0ij is equal to 1 if j was a customer of supplier i in the previousperiod (t − 1), and c0ij = 0 otherwise. Let C0 be the matrix with typicalelement c0ij .V0ij is a proxy to determine the compatibility between i’s industry andj’s industry. Specifically V0ij measures the proportion of firms in j’s industrythat were customers of a supplier in i’s industry, at time t− 1.41The function geoij is a categorical variable representing the firms’ geo-graphic proximity. Specifically, geoij is equal to one if the firms are in thesame geographic region (midwest, northeast, etc.), and zero otherwise.We also include a subset of idiosyncratic characteristics about the sup-plier, including the level of inventory turnover, financial leverage, size and40For example, the supplier may have inventory constraints, exclusivity agreements, orreputational concerns that prevent him from selling to a given customer.41When constructing this variable, we exclude the relationship between i and j. Thisallows us to avoid collinearity issues between c0,ij and V0,ij .109A.2. The Formation of Supply Chain NetworksFigure A.1: A sample network, where c12 = c32 = 1 and cij = 0 everywhereelse.age. All of these characteristics are measured at the beginning of the period.Finally, we assume that all the error terms, ηij , are iid and follow a logisticdistribution.Let α be the vector of coefficients from equation (A.1). If we define pijas the probability of observing a link between firms i and j, we thus havethatpij ≡ pr (cij = 1)= pr (Uj (i) > 0)=exp (α0 + αcc0,ij + αV V0,ij + αgeogeoij +∑k αkxki)1 + exp (α0 + αcc0,ij + αV V0,ij + αgeogeoij +∑k αkxki)where the last equality follows from the fact that the error terms, ηij , followa logistic distribution. Observe that pij is a function of α, and is conditionalon: (i) the state of the network at t− 1, C0, and; (ii) on the characteristicsof both firms, Xi and Xj . In other words, pij=pij(α|Xi,Xj,C0)Therefore, the likelihood of observing network matrix C, conditional onX and C0, is equal toLnetwork (α|C;X,C0)=∏i 6=j[pij (α|Xi,Xj,C0)cij × (1− pij ((α|Xi,Xj,C0)))1−cij]For example, consider the network from Figure A.1 and suppose that,given weights α, we have p12 = p31 = 0.3, p21 = p13 = 0.4 and p23 = p32 =0.8. We can verify that the likelihood of observing this network is Lnetwork =[(p12)c12 (1− p12)1−c12]×[(p13)c13 (1− p13)1−c13].... =[(0.3)1 (0.7)0]×[(0.4)0 (0.6)1]... = 0.0121.110A.2. The Formation of Supply Chain NetworksProcess 2: Determinants of ProductivityThe second process estimates the determinants of productivity: (i) the firmeffects, γ; (ii) the exogenous spillover effects, θ, and; (iii) the endogenouseffect, ω.Consider equation (2.2) and let LTFP (γ, θ, ω|TFP;C,X) be the likeli-hood of observing the productivity vector, TFP, given coefficient vectors(γ, θ, ω). This likelihood is conditional on: (i) network matrix C (which isformed in Process 1) and; (ii) the characteristics of the firms, X.Also, assume that the error terms from equation (2.2) are all nor-mally distributed with mean 0 and variance σ2. Under this assumption,we can verify that LTFP (γ, θ, ω|TFP;C,X) = φ (µTFP ,ΣTFP ), where φis the Probability Density Function of the Normal distribution, and whereµTFP = (I− ωW)−1 (γ + θW) and ΣTFP =σ2 (I− ωW)−1 (I− ωW′)−1.Estimation ProcedureAs explained above, we will jointly estimate: (i) the determinants of the se-lection process (α), and (ii) the determinants of productivity(γ, θ, ω), condi-tional on the selection process. Our goal here is to find the vector (α, γ, θ, ω)that maximizesL (α, γ, θ, ω|TFP,C,X,C0) =Lnetwork (α|C;C0,X)︸ ︷︷ ︸×selection process×LTFP (γ, θ, ω|TFP;X,C)︸ ︷︷ ︸TFP determinantsThis above estimation will allow us to control for partner selection biaseswhen estimating spillover effects. To estimate this function we use the sameapproach that can be found in the Appendix of Goldsmith-Pinkham andImbens (2013). This approach uses Monte-Carlo-Markov-Chain (MCMC)algorithms to estimate the distribution of the likelihood function.Data DescriptionGiven the size of our sample, firms are able to choose (literally) billions ofcombinations at the time of designing a supply chain structure. For thisreason, it is impossible to obtain results arising from a large sample. There-fore, we limit our estimation to a subsample of firms. In this subsample weonly include the most recent observations, i.e. those recorded between theyears 2006 and 2009. This subsample contains a panel of 1,697 firms andapproximately 5,600 linkages.111A.2. The Formation of Supply Chain NetworksVariable Coeff. t-statPrevious relationship (coij) 7.136*** 66.25Industry synergy (Voij) 8.825 *** 52.39Geographic Proximity (geoij) 0.50*** 10.29Supplier’s size 0.022*** 2.62Supplier’s age 0.009*** 6.63Supplier’s inventory turnover 0.112 1.57Supplier’s financial leverage 0.150* 1.82constant -4.95*** -97.92Linkages 5,605*** p<0.01, **p < 0.05,*p <0.1Table A.1: Model estimates for the selection equation.EstimatesIn Table A.1, we report estimates for the factors that influence a firm’sselection of partners (α). According to these estimates, there are two keypredictors in the formation of networks: the synergy between the industries,and the existence of a linkage in the previous year. From here we can seethat supply chains are largely determined by factors that are exogenous.We also find that a supplier is more likely to be selected by a firm if itis large, aged, or if the supplier is located in the same geographic region asthe customer. A supplier’s financial leverage and inventory control do seemto have a positive impact on the selection process, albeit this impact is notvery robust (i.e. it has a relatively small t−value).In Table A.2, we report estimates for the determinants of productivity.In Column 1, we report the Lee estimates without controlling for selectionprocess; in Column 2, we report the estimates with these controls.When we fail to control for the selection process, our estimates overre-port the magnitude of the endogenous effects. Specifically, we have that ωdecreases from 0.4826 to 0.3232.The exogenous spillover effects are not highly affected by the partnerselection biases. Note, however, that the spillover effect of inventory turnoveris negative in Column 2. Although this effect contradicts the estimates fromColumn 1, the effect is statistically insignificant.112A.2. The Formation of Supply Chain NetworksEffect Variable(1) (2)No Controls Selection controlsCoeff. t-stat Coeff. t-statEndogenous effect (ω) Cust. TFP 0.4826** (2.16) 0.3232** (1.98)Age 0.0185 (1.50) 0.2316 (0.60)Age2 -0.0003* (-1.72) -0.0004 (-0.69)Exogenous Size 0.0259 (1.01) 0.06195 (0.87)spillover Leverage -0.0082 (-1.35) -0.1122 (-1.14)effects (θ) Inv. Turnover 1.8121*** (2.65) -1.4732 (-1.44)Inv. Turnover2 -0.1561 (-0.55) -0.1040 (-0.71)Region: West 0.3879* (1.74) 0.2292 (1.45)Region: Midwest 0.4222*** (2.66) 0.1032 (0.8)Region: South 0.1966* (1.70) 01040 (1.04)Region Northeast -0.0162 (-0.11) 0.10273 (0.86)Age 0.0044 (0.49) 0.02118 (1.14)Age2 -0.0001 (-0.63) -0.0036 (-1.44)Size 0.0595*** (3.81) 0.03923** (2.05)Leverage -0.1097 (-1.11) -0.0105 (-0.22)Firm Inv. Turnover -1.6752* (-1.95) -1.8446 (-1.61)effects (γ) Inv. Turnover2 2.1655 (0.99) 1.3447 (1.61)Region: West 0.1900* (1.65) 0.5122 (0.87)Region: Midwest 0.0811 (0.71) 0.2357 (1.18)Region: South 0.0906 (0.97) 0.2234*** (2.70)Region Northeast 0.0786 (0.80) 0.0546 (0.49)Number of Observations 5,605 5,605*** p<0.01, **p < 0.05,*p <0.1Table A.2: Model estimates for the linear-in-means equation.113A.3. Variables Definition and ConstructionA.3 Variables Definition and Construction• Sales: Net sales [SALE in Compustat], deflated by the 4-digitindustry-level deflator. We obtain the deflators from the NBER-CESManufacturing Industry Database.• Capital: Gross property, plant and equipment [PPEGT], deflated bythe price deflator for investment. Since investment is made at vari-ous times in the past, it would be imprecise to use the current year’scapital deflator. To deflate capital stock, we use method adopted byBrynjolfsson and Hitt (2003). In this method, we deflate capital stockat the calculated average age of capital. To calculate the average age ofcapital stock, we divide accumulated depreciation [DPACT] by currentdepreciation [DP].• Materials: Total Expenses-labour expenses. Total expenses is equalto Sales minus operating income before depreciation and amortization.We obtain Operating Income from item 13 in compustat [OIBDP]. Theresulting value is deflated by the 4-digit deflator for materials• Labour expense: To proxy for labour expense we use sector-averagelabour cost per employee, and multiply it to the total number of em-ployees [EMP]. To determine the average sector labour cost, we use theannual sector-level wage data (salary plus benefits) from the Bureauof Labor Statistics. Labour expense is deflated by the price index fortotal labour compensation.• Value added: Sales (deflated) - Materials (deflated).• Capital investments (I): Capital expenditures, from Compustat[CAPX], deflated by the 4-digit industry-level deflator.• Firm age: Proxied by the year the firm first appeared in Compustat.• Firm size: Measured by the natural logarithm of total assets [AT],from Compustat.• Financial leverage: Measured as the ratio of total debt (short-termdebt [DLC] + long-term debt [DLTT]) to the book value of total assets.• Inventory turnover: Measured as the ratio of net sales to the levelof inventory [INVT].114A.3. Variables Definition and Construction• Geographic regions: Constructed by using the headquarters’ loca-tion of each firm. We looked at their home state [STATE] in Compu-stat. If this variable was not reported, we looked at their home city[CITY] to verify if the company is located in the U.S. or abroad.• Customer weight (wij) : Constructed as the ratio of the customersales [CSALE] to the sum of all customer sales in that year. Thisvariable was obtained from the Compustat business segments.115Appendix BProofs for Chapter 3Proof of Lemma 1: In stage 2, the integrated supply chain solvesmaxeS ,eO EX [Π (eS , eO, v,X)] = pi − ((1− v)xF (eS , eO) + P + eS + eO).By taking the first order conditions with respect to eO and eS we find that anycritical point must satisfyxβ(1−v)(e∗O+1)β+1(e∗S+1)1−β = 1 andx(1−β)(1−v)(e∗O+1)β(e∗S+1)2−β = 1.By solving for e∗O and e∗S , we find that e∗O =√x(1−v)β2−β(1−β)1−β− 1 and e∗S =√x(1−v)(1−β)1+βββ − 1 .To verify that this critical point is a maximum, we use the second or-der partial derivative test. We have that∂E[Π]∂e2S= − x(1−v)(1−β)(2−β)(eO+1)β(eS+1)3−β < 0,∂E[Π]∂e2O= − x(1−v)(β+1)β(eO+1)β+2(eS+1)1−β < 0 and∂E[Π]∂eO∂eS= − x(1−v)β(1−β)(eO+1)β+1(eS+1)2−β < 0,and∂E[Π]∂e2S∂E[Π]∂e2O−(∂E[Π]∂eO∂eS)2=[((x(1−v))(eO+1)β+1(eS+1)2−β)2(2β (1− β))]> 0.Hence, (e∗S , e∗O) is a maximum. Proof of Proposition 2: In stage 1, the supply chain solvesmaxv∈[0,1]EX[ΠFB (v,X)]subject to ΠFB (v, x) + W ≥ 0, whereΠFB (v,X)≡Π(eFBS (v) , eFBO (v) , v, P(v|eFBS (v) , eFBO (v)), X). By Lemma1 we can write EX[ΠFB (v,X)]=pi −(√xββ(1−β)1−β)(√(2−v)2(1−v))+ 2.Because ∂∂v√(2−v)21−v =√(2−v)21−vv2(2−v)(1−v) is positive for any v ∈[0, 1], it follows that ∂∂vE[ΠFB (v, x)]< 0 for any v ∈ [0, 1]. Therefore,arg maxv∈[0,1]{EX[ΠFB (v,X)]}= 0. In other words, v = 0 maximizesthe (unconstrained) profits for the centralized supply chain and, hence, thesupply chain chooses v = 0 whenever ΠFB (0, x) +W ≥ 0.Now suppose that ΠFB (0, x) +W < 0. Here, v = 0 is not in the feasibleregion. Furthermore, for any x ≥ 1,42 it is true that ∂∂vΠFB (v, x) |v=0=√x(√x−√14(1−β)1−βββ)> 0, which implies that the wealth constraint is42Note that because we assume that x must be large to guarantee an interior solutionfor the effort levels, it can be shown that x ≥ 1.116Appendix B. Proofs for Chapter 3relaxed when v is increased beyond 0. Using this fact in conjunction withthe fact that ∂∂vEX[ΠFB (v,X)]< 0, we find that the optimum is given byv = vFB where ΠFB(vFB, x)+W = 0. Proof of Lemma 3: In stage 2, the best response level of effort forthe supplier i→s equal to e˜S |eO = arg maxeS≥0EX [ΠS (eS , w, y, P,X|eO)]=(xy(1−β)(eO+1)β) 12−β− 1, and the best response function for the operator ise˜O|eS =EX [ΠO (eO, w, y, v, P,X|eS)]=(xβ(1−y−v)(eS+1)1−β) 1β+1− 1. The Nash equi-librium effort levels is given by the fixed point correspondences of thebest response functions, i.e. e∗S≡e˜S |e˜O and e∗O≡e˜O|e˜S , where e∗O (y, v) =√x(β(1−y−v))2−β((1−β)y)1−β− 1 and e∗S (y, v) =√x(y(1−β))1+β(β(1−y−v))β− 1.Proof of Lemma 4: This Lemma follows di-rectly from Lemma 3, by solving explicitly forF (e∗S (y, v) , e∗O (y, v)) =[(1 + e∗O (y, v))β (1 + e∗S (y, v))1−β]−1=[√x(β (1− y − v)β)((1− β) y)1−β]−1Proof of Proposition 5: We prove our results for the case where β 6=0.5, as the proof for the case where β = 0.5 is identical. In region UC,the operator seeks to find maxw,y,v EX [ΠO (e∗O, w, y, v, P (v|e∗S , e∗O) , X) |e∗S ]subject to (IR), y ∈ [0, 1] and v ∈ [0, 1− y],e∗S = arg maxeS≥0 {ΠS}. First,note that if IR is non-binding and (w∗, y∗, v∗) are maximizers, then w∗ >xy∗Φ (y∗, v∗) + e∗S . But if this were true, the operator could decrease w∗by ε =w−xy∗Φ(y∗,v∗)−e∗S2 , which increases EX [ΠO] without violating anyconstraint. Therefore, w∗ = xy∗Φ (y∗, v∗) + e∗S , which implies that IR isbinding.If we plug w∗ = xy∗Φ (y∗, v∗) + e∗S into the objective function, we canre-write the operator’s problem as E [Π∗O (y, v)] =x(y+β−vβ−2yβ+1)Φ(y,v) + 2 sub-ject to y ∈ [0, 1] and v ∈ [0, 1− y]. But note that limy→0 Π∗O (y, v) =limy→1 Π∗O (y, v)= -∞ and limv→1−v Π∗O (y, v) = −∞. Hence the optimalsolution must be such that y ∈ (0, 1) and v ∈ [0, 1 − y). Under this as-sumption, we write the Lagrangian function of the operator’s problem asL (y, v) ≡ xΦ (y, v) + e∗S (y, v) + e∗O (y, v) − λv, where λ is the Lagrangianmultiplier for the non-negativity constraint, v ≥ 0. In this program, the KKTconditions are equal to xΦy+e∗0,y+e∗S,y = 0,xΦv+e∗0,v+e∗S,v = λ,λ ≥ 0,v ≥ 0and vλ = 0.43 Through algebraic arrangements, we can show that if v 6= 0,43Where, to simplify notation, we let Φy ≡∂Φ(y,v)∂y ,e∗S,y ≡∂e∗S(y,v)∂y ,etc.117Appendix B. Proofs for Chapter 3the KKT conditions are only solved when v = −β − 1 and y = β + 1, whichviolates the non-negativity of v. Therefore, the KKT conditions can only besolved when v∗ = 0 and y∗ =(1+β)(β(1+β)−2+√β(8−7β−2β2+β3))2((1+β)2−2).To verify that y∗ is a maximum over the region where v = 0, wecheck the second order condition:∂2E[ΠO(y,0)]∂y∂y =−Φ(y,0)xh(y,β)4y2(1−y)2, whereh (y, β)≡β (2− β) y3−(β3 + 2β2 − 2β − 3)y2+(β3 − 6β2 − β + 6)(1− y)−3(1− β2)−3(1− β2). Note that h (y, β) is positive because: (i)arg miny∈[0,1],β∈[0,1] h (y, β) = {(1, 0) , (0, 1)} and; (ii) h (0, 1) = h (1, 0) = 0.Hence,∂2E[ΠO(y,0)]∂y∂y =−Φ(y,0)xh(y,β)4y2(1−y)2< 0.Proof of Proposition 6: Consider the operator’sproblem in region SC. Here, the operator must findmaxw,y,v EX [ΠO (e∗O, w, y, v, P (v|e∗S , e∗O) , X) |e∗S ] subject to constraintse∗S , v ∈ [0, 1− y] and y ∈ [0, 1]. By the same argument as the one made inthe Proof of Lemma 5, we can show that y∗ ∈ (0, 1) and v∗ ∈ [0, 1− y).By Lemma 5, we can show that the ex post profits forthe supplier in case UC, given an operational failure, areΠS(e∗O(yUC , 0), wUC , yUC , 0, x|eS(yUC , 0))= yUCx(1− ΦUC). Now, by defi-nition ofWCS , we know that ifWS +ΠS = WS +yUCx(1− ΦUC)≡W IS > 0,then WCS is not binding and, therefore, the optimal solution must be givenby vSC = 0,wSC = wUC and ySC = yUC .Now, assume that WS < W IS or, alternatively, that WCSbinds at optimum. Therefore, by this constraint we know thatw∗ = yx + e∗S (y, v) − WS . We can plug w∗ to the ob-jective function and re-write the Lagrangian of operator’s problemas L(w, y, v, λ, µ) ≡pi−(xΦ (y, v) (1− y) + xy + e∗O (y, v) + e∗S (y, v) +WS)−vλ − µ (WS − xy (1− Φ (y, v))). Here µ and λ are the shadow price forIR and for the non-negativity constraint for v. The KKT conditionsgive us xΦ − xΦy (1− y) − x − e∗O,y − e∗S,y − µx ((1− Φ)− yΦy) = 0,xΦv (1− y) + e∗O,v + e∗S,v − λ + µ (xyΦv) = 0, µ (xy (1− Φ)−WM ) = 0,vλ = 0, µ ≥ 0, v ≥ 0 and λ ≥ 0.We can re-write these conditions to showthat λ = xβ((1−Φ)(y+v)−(1−β)(1−v−2y))( 2Φ−(1+β))(1−y−v)−yβand µ=(2xβ((1−Φ)(y+v)−(1−β)(1−v−2y))( 2Φ−(1+β))(1−y−v)−yβ)((1−y−v)−xΦβ2 (1−yβ−(2−β)(1−y−v))yxΦβ). Byrearranging these conditions, we can see that when WS > W IIS ≡x (1− β)(1− 2yUC)but WS ≤ W IS = yUCx(1− ΦUC), the KKT solutionscan only be solved at a point where λ > 0 and µ > 0, i.e. when both the IR118Appendix B. Proofs for Chapter 3and non-negativity constraints are binding. The optimal solution is thusgiven by v = 0 and ySC , where ySCsolves (1− Φ (y, 0)) = WSxy .Now, we can also check that when WS ≤ W IIS , the non-negativityconstraint is non-binding, i.e. λ = 0 and v > 0. Here, ifWS ≥ xm(1− m(2−β)1−β(1−m)), where m solves x (1− β)m((1−mβ)β(2−β)m(1−β))β=((1+mβ−β)(2−β)m)2, the individual rationality constraint is binding and, atoptimum vSC = ySC(x(1−ySC)(1−β)WS+xySC(1−β)− 1)> 0 where ySC satisfiesySC(1− ΦSC)= WSx . Finally, if WS < xm(1− m(2−β)1−β(1−m)), the IR con-straint is non-binding (i.e. µ = 0). Therefore, vSC = 2(1−β)(1−2ySC)2−β > 0 andySC satisfying ΦSC = ySC(2−β)1−β(1−ySC) . Proof of Proposition 7: In region OC the operator seeks tofind max EX [ΠO (e∗O, w, y, v, P (v|e∗S , e∗O) , X) |e∗S ] subject to IR, WCO,y ∈ [0, 1] and v ∈ [0, 1− y]. By Lemma 5, the ex post prof-its for the operator (in region UC), given the realization of an oper-ational failure, are equal to ΠO= 2 + pi − xΦUC(2yUC (1− β) + β)−x(1− yUC). Hence, constraint WCO is non-binding when WO > W IO ≡(2 + pi − xΦUC(2yUC (1− β) + β)− x(1− yUC)), which implies that the op-erator’s problem is identical to the one in case UC, i.e. that vOC = 0 andyOC = yUC .Now, assume that WO ≤ W IO, i.e. that WCO binds at optimum. Asa result, we can re-write ΠO (eO, w, y, v, P, x) + WO = 0. Moreover, by anargument similar to the one made in the Proof of Lemma 5, we have that y∗∈ (0, 1), v∗ ∈ [0, 1− y) and IR is binding. As a result, we can re-write theoperator’s problem as maxy,v pi − xΦ (y, v) − e∗O (y, v) − e∗S (y, v) subject toΦ (y, v)x (y + v)+x (1− y − v) + e∗O (y, v) + e∗S (y, v)− pi = WO and v ≥ 0.If we let µ and λ be the shadow prices for con-straint WCO and for the non-negativity constraint ofv, we can re-arrange the KKT conditions to show thatλ=(2(1−β)β (1−y−v)2)((1−y)(1−Φ)+β(1−v−2y))((v−1)(v+2y−1)β2+(v(3−y)−2(v2+y2)+y−1)β+(v2−v(1+y)+2y(1−y)))Φ−2y(1−v−y) .By using this condition, jointly with the conditions that v ≥ 0, λ ≥ 0,λv = 0 and with the fact that WCO is binding, we find that when WO islarger than W IIO ≡ xΦ (m, 0) (2m (1− β) + β) + x (1−m) − 2 − pi, wherem solvesx((1−m)β)β(1+β( 1−2m1−m ))2(m(1−β))β−1= 1, then vOC = 0 and yOC satisfies2 + WO = xΦOC (2 (1− β) + β) + x(1− yOC). However, we can also verify119Appendix B. Proofs for Chapter 3that when WO ≤W IIO , then the KKT solutions are solved by setting λ = 0,vOC =(11−β)(y22 −(1−y)(WO+2−xy)(1−β)−xxβ) 12−β + 2y(34 + β)and yOCsatisfying ΦOC =β(1−vOC−yOC)1−yOC + 1.Proof of Proposition 8: Suppose that WS ≤ ∞ andWO ≤ ∞ . In this case, the operator’s problem is to findmaxw,y,vEX [ΠO (e∗O, w, y, v, P (v|e∗S , e∗O) , X) |e∗S ] subject to IR,WCS ,WCO,y ∈ [0, 1] and v ∈ [0, 1− y] . By Propositions 6 and 7, we know that ifWS > W IS and WO > WIO then the WCS and WCO constraints are non-binding. As a result, the operator’s problem reduces to the one analyzed incase UC.Suppose that WS ≤ W IS . By Proposition 6, we know that WCS con-straint is binding. Moreover, we can verify that, in this sub-region, theWCO constraint is binding if and only if WO < W˜(ySC , vSC)− WS . Ifthis constraint is non-binding, the operator’s problem is identical to theone analyzed in case SC. If, however, WO ≤ W˜(ySC , vSC)−WS , the so-lution to the operator’s problem can be found by solving for the wealthconstraints of the operator and the supplier. This means that the opti-mal solution must ensure that WO + WS = W˜ (y∗, v∗) and 2 + pi + WO =x (1− y∗ − v∗) +xΦ (y∗, v∗) (2y∗ + v∗ + β (1− 2y∗ − v∗)). We can make thesame analysis for the case where WO ≤ W IO to derive the remaining resultsin the proposition. Proof of Lemma 9: Suppose the effort of the parties is sequen-tial, and that the supplier leads. Under these assumptions, thebest response level of effort for the operator, given eS , is equal toe˜O,seq (eS) =arg maxeO≥0 EX [ΠO (eO, w, y, v, P,X|eS)]=(xβ(1−y−v)(eS+1)1−β) 1β+1−1. As such, the optimal level of effort for the supplier is e∗S,seq =arg maxeS≥0EX [ΠS (eS , w, y, P,X|e˜O,seq)] =√x(β(1−y−v))2−β((1−β1+β)y)1−β − 1. By plug-ging e∗S,seq into the operator’s effort function, we can verify that e∗O,seq =e˜O,seq(e∗S,seq)=√x(β(1−y−v))2−β((1−β1+β)y)1−β − 1.Proof of Proposition 10: Suppose that both the supplier and the operatorhave the option of purchasing insurance coverage, vS and vO. In this model,the supplier’s and the operator’s ex post profits are ΠS (eS , w, y, vS , PS , X) =T (w, y,X) − IS (vS , PS , X) − eS and ΠO (eO, w, y, vO, PO, X) =pi − X − T (w, y,X) − IO (vO, PO, X) − eO, where IS (vS , PS , X),IO (vO, PO, X) and T (w, y,X) are defined as in Section 3. In this case,120Appendix B. Proofs for Chapter 3the best response level of effort for the supplier and operator are equal toe˜S |eO= arg maxeS≥0{w − (y−vS)x(eS+1)1−β(eO+1)β − eS}=(x(y−vS)(1−β)(eO+1)β) 12−β− 1,and e˜O|eS= arg maxeO≥0{pi − w − x(1−y−vO)(eS+1)1−β(eO+1)β − P − eO}=(xβ(1−y−vO)(eS+1)1−β) 1β+1− 1. By solving for the correspondences of these functions,we get that e∗O (y, vS , vO) =√x(β(1−y−vO))2−β((1−β)(y−vS))1−β − 1 and e∗S (y, vS , vO) =√x((y−vS)(1−β))1+β(β(1−y−vO))β − 1. Now, note that the optimal level of coverage for thesupplier are given by v∗S = arg maxvS≥0EX [ΠS (e∗S , w, y, vS , PS , X) |e∗O] =w − (y−vS)x(e∗O+1)β(e∗S+1)1−β − P (vS |e∗O, e∗S) − e∗S . By plugging the expression ofe∗S and e∗O in the expected profits of the supplier, and by performing basicoptimization techniques, we can verify that v∗S =βy(1+β) . To prove part 2 ofthe proposition, we first need to write e∗S(y, βy1+β , vO)=√x(β(1−β)1+β y)1+β(β(1−y−vO))βand e∗O(y, βy1+β , vO)=√x(β(1−y−vO))2−β(β(1−β)1+β y)1−β − 1. Using these results, we candirectly verify that the ratioe∗O+1e∗S+1is identical to the ratio given in equation(3.3) 121Appendix CProofs for Chapter 4Proof of Lemma 11First, note that, from equation (4.2) the stage-3 profits for type-τ firmare equal to Πτii (rˆτii |X, s, b) = ρX − s − mτ . This allows us to re-writethe stage-2 profits, which are given in equation (4.3), as Π˜τii (qi, qj |s, b)= qi [qjρD + (1− qj) ρM −mτi + s] and therefore, to re-writeΠ∗i(qτii , qHj , qLj |s, b, p)= qτii[ρD(pqHj + (1− p) qLj)]+qτii[ρM(p(1− qHj)+ (1− p)(1− qLj))]+ s−mτiRecall that the best response function for firm i, with type τ, is given byBτi(qHj , qLj |s, b, p)= arg maxqτii ∈[0,1]{Π∗i(qτii , qHj , qLj |s, b, p)}.A Bayesian Nash Equilibrium exists if and only if there is a type-contingent strategy qˆτ for τ = H,L, such that Bτi(qˆH , qˆL|s, b, p)= qˆτ . Inother words, to find a BNE, we need to find a set of fixed points qˆH andqˆL for the best response functions of high- and low-ability firms. By writingthese two conditions explicitly, one can show that a BNE equilibrium existsif we can find qˆH and qˆL, such that the following system is solved:qˆH = max{min{(s+ ρM −mH)p (ρM − ρD)−1− ppqˆL, 1}, 0}qˆL = max{min{s+ ρM −mL(1− p) (ρM − ρD)−p1− pqˆH , 1}, 0}To find the BNE for all regions, we need to solve for these equations. Wecan find a solution by first finding the conditions under which(qˆH , qˆL)=(1, 1) and(qˆH , qˆL)= (0, 0). That is, we can verify that(qˆL, qˆH)= (0, 0) ifand only if s + ρM −mH < 0. We can also verify that(qˆL, qˆH)= (1, 1) ifand only if s+ ρD −mL ≥ 0. We can solve the remaining regions by movingaway from these extreme thresholds. 122Appendix C. Proofs for Chapter 4Proof of Lemma 12First, let αH and αL be arbitrarily close to 0. By looking at equation (4.2) ,jointly with equations (4.5) and (4.6) , we can see that (for any (s, b)) thefollowing two conditions are satisfied: (i)Wτi (M, s, b) < ρM+cM−d and (ii)Wτ1,τ2 (D, s, b) < 2ρD + cD − 2d. Moreover, recall that by the assumptionsstated at the end of §3, we have that ρM +cM −d < 0 and 2ρD+cD−2d < 0. This implies that, whenever αH and αL are very close to 0, both themonopoly and the duopoly welfare will be negative, regardless of the typeof firm(s) that enter the market. In other words, it is socially optimal tohave no firms operating in the market. As a result, the policy (0, 0) willmaximize the expected social welfare. Moreover, we can also see that becauseρD < ρM < d, then the firms will never profit from entering the market and,at equilibrium, both high- and low-ability firms will choose qi = 0.Second, let αH and αL be arbitrarily large. This implies that for anypolicy (s, b) , all firms will choose qi = 1. This is true because if ατ → ∞,then mτ → 0 and, therefore, Πτii (rˆτ |M, s, b) > Πτii (rˆτ |D, s, b) = ρD + s −mτ > 0. In other words, the firms will always profit from entering to themarket. The result follows immediately.Third, let αH be arbitrarily large, but αL be arbitrarily close to 0. Bythe argument above, the high ability firms will always choose qi = 1. Also,because αL is arbitrarily close to 0, then a low-ability (if it enters) will causean accident with probability arbitrarily close to 1. By the assumptions statedat the end of §3, then the social planner will want to induce market exit bylow-ability firms. Hence, by Lemma 1, the optimal policy (s∗, b∗) must besuch thats∗ + pρD + (1− p) ρD − (1− b∗) d+αL ((1− b∗) d)22≤ 0 (L1.1)i.e., such that low-ability firms will choose qi = 0. Otherwise, the socialplanner could always decrease the level of incentives to satisfy equality in(L1.1). By doing this, he would decrease the entry probability of low-abilityfirms, without decreasing the entry decision of high-ability firms.By the above arguments, there must exist α1 and α2 satisfying theproperties of Lemma 2. Proof of Proposition 13To decrease the notational burden in the proof, we will simplify the notation.We let, for τ = H,L123Appendix C. Proofs for Chapter 4Φτ (b) = d (1− ατ (1− b) d) +(ατ (1− b) d)22ατφτ (b) = λ (1− ατ (1− b) d) bγ (κ) = 2(cM −cD (κ)2+ (1 + λ) (ρM − ρD (κ)))Here, Φτ (b) > 0 refers to the expected primary accident costs, as afunction of ex post liability protection. φτ (b) > 0 refers to the opportunitycost of providing ex post subsidies b. γ (κ) > 0 is a measure of the expectedwelfare gains as a function of the competition parameter, κ.1. Suppose that αH ≥ α1 and αL ≤ α2. By Lemma 2, we know that theoptimal policy will be such that (s∗, b∗) will induce entry by (only)high-ability firms. By Lemma 1, therefore, we know that (s∗, b∗) issuch thats∗ + pρD + (1− p∗) ρD −mL ≤ 0 (P1.1)where mL = (1− b∗) d +αL((1−b∗)d)22 . Now, if we take the first ordercondition of E [W (s, b)] with respect to s, we will find that whenever(s∗, b∗) satisfies condition (P1, 1) , thenpqˆH (s, b) = −ΦH − cM − ρM + λ (s+ φH)γmust be satisfied. Similarly, if we take the first order condition withrespect to b, we will find thatpqˆH (s, b) =ΦH + λ (s+ φH)− p (cM + ρM )2cM + 2ρM − cD − 2ρD − (ρM − ρD)Φ′H+λφ′−Φ′H+φ′must also be satisfied. By combining these conditions, we find thatthe policy that satisfies the first order conditions must also sat-isfy λ =Φ′H+λφ′H−Φ′H+φ′Hor Φ′H (λ+ 1) = 0, where Φ′H (b) = bd2αH .Hence, at the first order condition, it must be true that b∗ =0. Furthermore, if we solve for s (after setting b = 0) wefind that the optimal level of ex ante subsidies satisfies sH =124Appendix C. Proofs for Chapter 42√dα − ρM + (ρM − ρD) min(cM+(λ+1)(ρM−d+αHd22))(2cM−cD+2(1+λ)(ρM−ρD)), p, where(cM+(λ+1)(ρM−d+αHd22))(2cM−cD+2(1+λ)(ρM−ρD))= p whenever qˆH (s∗, b∗) = 1.Now, to show that the first order condition is sufficient, we use the sec-ond order partial derivative test. Note that∂2E[W(sH,0)]∂s2 =2γ(ρM−ρD)2 <0,∂2E[W(sH,0)]∂s∂b =2γ(1−dαH)(ρM−ρD)2 and∂2E[W(sH,0)]∂b2 =(1−dαH)(−(γ+(ρM−ρD)(1+λ))sH+(ρM−ρD)cM−ρMγ)(ρM−ρD)2 . Hence, thedeterminant of the Hessian matrix will be equal to2γd(1−dαH)(1+λ)(ρM−ρD)2((cM−(λ+1)ρM−d)γ)> 0. By the second order par-tial derivative test, we conclude that (s∗, b∗) globally maximizesE [W (s, b)] .2. Suppose that αH ≥ α1 and αL ≥ α2. By Lemma 2, we have that(s∗, b∗) induces entry by high- and low-ability firms. By Lemma 1,therefore, we have that the optimal policy will be one in have thats∗ + pρD + (1− ρD) ρM − d (1− b∗) +αL (d (1− b∗))22> 0 (P1.2)Now, if we take the first order condition with respectto s, we will be able to see that if (P1.2) is satisfied,the optimal solution will also satisfy ΦL + λ (s+ φL) −((cM + ρM ) (1− 2p) + 2 (cD + 2ρD)) + (ρM − ρD)λ =qˆL (s, b) (1− p) [cD + 2ρD − 2cM − 2ρM − (ρM − ρD)λ] . By using theexplicit form of qˆL (s, b) , we will be able to see that the optimal level ofex ante subsidies is given by s∗ =sHL ≡(1− bHL)d−αL((1−bHL)d)22 −ρM+(ρM−ρD)[cM+(1+λ)(ρM−d+αL(1−bHL)d2(1− 1−bHL2))]2(1+λ)(ρM−ρD(κ))+(2cM−cD(κ)).To ver-ify that this first order condition is a maximum, note that∂2E[W(s,b)|p](∂s)2=(−2γ(ρM−ρD)2)< 0.Now that we have solved for s∗, we proceed to solve for the optimal level ofb. To do this, we can plug in sHL in the welfare function, and see thatW(sHL (b) , b)≡ WHL (b) =((cM+(λ+1)(ρM−d+αL(1−b)d2(1− 1−b2 )))2)γ ++bd2(αH−αL)(b+2(λ+1)b)pγγ . By taking the first order condition with re-125Appendix C. Proofs for Chapter 4spect to b, we will get that ∂∂bWHL (b) = 0 whenever b satisfiesp(αH−αL)αL(λ+1)(1 + λ(1−2(1−b))b)=−cM+(λ+1)(ρM−d+αL(1−b)d2(1− 1−b2 ))γ We can re-write this equation asA2(1− bHL)3− 3A2(1− bHL)2+ (A−M − T )(1− bHL)+ M = 0, whereA = d2 (1 + λ)αL > 0, M = cM + (1 + λ) (ρM − d) +pγ(αH−αL)αLand T = p(αH−αL)γλαL(λ+1) > 0.Through the cubic formula, we findthat the only real root that solves this equation is given at bHL =1A∑2i=1[T + (−1)i(T 2 −(2(M+T )+A3)327A) 12] 13> 0. Note that the inequal-ity follows directly from the fact that T > 0. To verify that this point is amaximum, we evaluate∂2WHL(b)∂b2 |b=bHL , and find that∂2WHL(bHL)∂b2=2d2αLγT −∑i=1,2T + (−1)i(T 2 −(2 (M + T ) +A3)327A) 12133<−2d2αLTγ< 0where the first inequality follows from the triangle inequality for the p-norm for p = 13 .Proof of Lemma 14One can verify, by plugging (s∗, b∗) into the optimal entry strategies, thatqˆL (s∗, b∗) = max(cM+(λ+1)(ρM−d+αL(1−b∗)d2(1−(1−b∗)2)))(2cM−cD(κ)+2(1+λ)(ρM−ρD(κ)))− p1− p, 0where qˆL (s∗, b∗) > 0 iff the optimal policy lies in regionHL, and qˆL (s∗, b∗) =0 iff the optimal policy lies in region H.126Appendix C. Proofs for Chapter 4Note that in region HL, the optimal level of ex post subsidies, b∗ = bHL,is a function of κ, where∂b∗ (κ)∂κ= −∂γ(κ)∂κ(αLd2(1− b∗2)+ 2 (cM + (λ+ 1) (ρM − d)))2γ (κ)2(d2αLb∗γ(κ) −pλ(αH−aL)λ1αLb∗2)where γ is defined as in the beginning of Proposition 1. Hence, if we usethe chain rule to differentiate qˆL with respect to κ , we will find out that∂qˆL∂κ= −3d2 (1 + λ)αL∂γ(κ)∂κ2γ2 (1− p)where γ′ (κ) = −c′D (κ) − 2 (1 + λ) (ρ′D (κ)). The result follows directly,by observing that∂qˆL∂κ ≥ 0 iff∂γ(κ)∂κ ≤ 0 and vice versa. Proof of Proposition 15This result follows directly from Lemma 3, by noting that the welfare gains,given more aggressive competition, are given byc′(κ)2ρ′D(κ)− 1. Hence, whenc′(κ)2ρ′D(κ)− 1 is smaller than 0, qˆL will decrease (and vice versa). Moreover, ifqˆL decreases to 0, then the optimal policy will lie in region H, and the socialplanner will not offer ex ante subsidies. Proof of Proposition 16To show this result, we use an argument similar to the one we used to proveLemma 2. First, we show that, in region HL, ∂b∗∂d =dαL(1−b2)γ(d2αLbγ −pλ(αH−αL)λ1αLb2) .Hence, we can use the chain rule (by noting that b∗ is a function of d) todifferentiate qˆL with respect to d. By Lemma 1 and Proposition 1, we obtainthat∂qˆL∂d= max{−d (λ+ 1)2 αL(1− b2)(cM + (λ+ 1) ρM )γ (1− p) (αH − αL), 0}From the above result, we will be able to see that qˆL is decreasing ind. Hence, there must exist a d large enough so that, evaluated at such d, itmust be true that qˆL = 0. By Lemma 2 and Proposition 1, the result willfollow. 127Appendix C. Proofs for Chapter 4Proof of Proposition 17To show this result, first note that (in region HL) wehave that ∂b∗∂λ =(p(αH−αL)(1−b2)γ(cM+(1+λ)d)αL(λ+1)2γ2)> 0, ∂s∗∂λ =−(cM+(1+λ)(ρM+αHd22))2(αH−αL)γαL< 0. Now, if plug the optimal pol-icy into the equilibrium entry strategies, we find that∂qˆL(s∗(λ),b∗(λ))∂λ =− cDαLd2+γ+(ρM−ρD)d2(1−p)γ2 < 0. This means that the level of ex ante subsidieswill be decreasing as λ increases. Conversely, we see that the level of ex postsubsidies will increase as λ increases. However, when λ moves beyond somethreshold, then qˆL will decrease to the point where the low-ability firms willexit the market. Here, the level of ex post subsidies will decrease to 0, ands∗ will eventually converge to sH. 128


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