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Essays in policy analysis and strategy: entrepreneurship, joint venturing, and trade Arend, Richard James 1995

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ESSAYS IN POLICY ANALYSIS AN]) STRATEGY:ENTREPRENEURSHIP, JO1I%JT VENTURING, AN]) TRADEbyRICHARD JAMES ARENI)B.A.Sc., The University of Toronto, 1986M.B.A. (Honours), York University, 1989A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIES(Commerce and Business Administration)We accept this thesis as conformingto the rquired standardTHE UNIVERSITY OF BRITISH COLUMBIAJuly 1995° Richard James Arend, 1995In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of (vvc€. (PA/ ck\f >-The University of British ColumbiaVancouver, CanadaDateDE-6 (2/88)ABSTRACTSeparate essays on entrepreneurship, joint venturing, and trade comprise this thesis.The emergence of entrepreneurship is common in the real world but relatively less so in classicaleconomic models. If industry incumbents are attributed with full rationality and perfect foresight,then there are few, if any, profitable opportunities left for new entrants (entrepreneurs) toexploit. This essay explains how entrepreneurs can emerge in a dynamic world when firms mustchoose between a technology strategy that is either statically or dynamically efficient. A modelis developed which shows how such opportunities for new entry can occur when incumbents arecaught in a Prisoners’ Dilemma game involving technology strategy. A relevance measure andpolicy implications are then explored.Joint ventures, especially of the R&D type, are becoming increasingly important as a way togain needed technological and market competencies. Unfortunately, many joint ventures havethe characteristics of a Prisoners’ Dilemma. Firms may cooperate or defect in the venture. Ifcontracts, side-payments, and third-party verification of the venture outcome are unavailable,then the dominant solution to the Prisoners’ Dilemma (mutual defection) results. This paperproposes the use of an ex-ante auction to obtain a Pareto-improvement for these ventures. APareto-improvement is assured when non-transferable costs and benefits of firms are notconditional on joint venture strategies. When this condition is not met restrictions are requiredto obtain the Pareto-improvement.iiThe problem of trade between countries that share an international open access resource isbecoming significant as the world reaches the limits of critical shared resource stocks. It ismodelled as a world with one primary factor, two intermediate goods, one final good (harvestedfrom the open access resource), and two nations where it is assumed that either the trading takesplace over one stage (nations are price-takers), or two stages (nations have market power).Imperfect competition and open access generated externalities affect the trading efficiency. Tomaximize world welfare this essay recommends subsidizing R&D where comparative advantageexists, and creating international agreements to ensure the one-stage game structure is used whentrading.111TABLE OF CONTENTSAbstractTable of Contents ivList of Tables viiList of Figures viiiAcknowledgement ixINTRODUCTION 1Chapter One Essay One:Technological Force: the Emergence of EntrepreneurshipIntroduction 6The Model 10Important Definitions 10Definition of the Game 13Analysis of a Specific Game 21Considering an Extension to the Game 32Conclusions 35Chapter Two Essay Two:An Auction Solution to the Joint Venture Prisoners’ DilemmaIntroduction 38Description of an Application 43The Game 50The Basic Application 52An Application with n >2 Firms 60An Application with Asymmetric Firms 61An Application with StrategicNon-Transferable Costs (SNTCs) 69Solutions Under General SNTCs 76The New Auction Solution 78The Contracting Alternatives 80Conclusions 84ivChapter Three Essay Three:Fish and Ships: Trade with Imperfect Competition andan International Open Access ResourceIntroduction 87The Assumptions, the Model and the Autarky Case 91Analysis of the Trade in a Ricardian World- The One Stage Game 98Case i. One Stage Game with Unrestricted Stocks 98Case ii. One Stage Game with Restricted Stocks 108Analysis of Trade with Production Precommitments- The Two Stage Game 111Case iii. Two Stage Game with Unrestricted Stocks 113Case iv. Two Stage Game with Restricted Stocks 125Summary and Conclusions 132Chapter Four Overall Conclusions 135Bibliography 141Appendix Legend for Appendices One to Six 146Appendix One Payoff Requirements Under Game Regimes 147Appendix Two Numerical Examples of the Normal Form Game 150Appendix Three First Period Entry Restrictions 151Appendix Four Division of Parameter Space by Game-Type 153Appendix Five Accounting of Possible Games 155Appendix Six Existence of Prisoners’ Dilemma 160Appendix Seven Table of Solutions 162Appendix Eight The Coin Flip Solution 163Appendix Nine Alternative Solutions Under Special Futures Contracts 166Appendix Ten Analysis of SNTC Solutions and Some Extensions 169Appendix Eleven Equilibria with Restricted Bidding 174vAppendix Twelve The Nash Bargaining Solution 176Appendix Thirteen Analysis of Case iii. Under Stackelberg Leadership 180Appendix Fourteen Proof of Specialization in One-Stage Gamewith Unrestricted Stocks 183Appendix Fifteen Proof that Specialization in the One-Stage Gameis Greater than that in the Two-Stage Game 185Appendix Sixteen Concluding Proposition of Chapter Three 188viLIST OF TABLESTable One Bids and Payoffs for the Auction 57Table Two Comparative Statics of the One-Stage Gamewith Unrestricted Stocks 105Table Three Comparative Statics of the Two-Stage Gamewith Unrestricted Stocks 120Table Four Table of Solutions to the JVPD 162Table Five Payoffs under the Coin Flip Solution 165viiLIST OF FIGURESFigure One Time Line of the Model 14Figure Two The Normal Form of the Model 16Figure Three Prisoners’ Dilemma Payoff Ordering Requirements 24Figure Four The Auction Solution Time Line 54Figure Five Time Line of the Two Stage Game 112Figure Six Effects of Unilateral Specialization on Elasticity 123Figure Seven Numerical Examples of the Normal Form Game 150Figure Eight Effect of Changes in Elasticity onParameter Space Division 153Figure Nine Effect of Changes in Fixed Costs onParameter Space Division 154Figure Ten Stackelberg Leadership in the Two-Stage Game 182viiiACKNOWLEDGEMENTFinancial support from SSHRC and U.B.C. is gratefully acknowledged. I thank my supervisorDr. Thomas Ross, and the rest of my committee Dr. Raphael Amit and Dr. William Strange.I also thank Dr. James Brander, Dr. Kenneth MacCrimmon, and others in the Policy Analysisand Strategy Division, as well as all participants in my seminars at U.B.C. for their feedbackand input on this paper. I am very thankful to my family, to Ruth, and to my friends for theirpatience and support throughout this process. A special thanks goes to the champion CommerceSoftball Team for providing me a healthy alternative to work.ixINTRODUCTIONThese three essays in policy analysis and strategy explore separate and important economicissues in a similar way. First, a theoretical approach is used to investigate specific aspects ofentrepreneurial emergence, of inefficiencies in joint ventures, and of trade involving aninternational open access resource. Standard, and in most cases, classical economic and noncooperative game theoretical assumptions are applied wherever possible in the modelling of theissues. Second, each issue entails an effect on, or by, technology. Third, each analysis providessome practical policy recommendations. And fourth, each essay provides analysis in somedynamic context.The first essay, entitled “Force: The Emergence ofEntrepreneurship”, explainshow entrepreneurs can beat capable incumbents when new opportunities arise. The incumbentsrationally decide to let the entrepreneurs beat them. This explanation and the focus on theincumbents rather than on the entrepreneurs are not the only ways in which this paper differsfrom the other literature.The emergence of entrepreneurship is common in the real world but relatively less so in classicaleconomic literature. If industry incumbents are attributed with full rationality and perfectforesight, then there are few, if any, profitable opportunities left for new entrants (entrepreneurs)to exploit. This essay explains how entrepreneurs can emerge in a dynamic world when firmsmust choose between a technology strategy that is either statically or dynamically efficient. A1model is developed which shows how such opportunities for new entry can occur whenincumbents are caught in a Prisoners’ Dilemma game involving technology strategy.The model provides an empirically testable explanation of how industry structure changes whenany technological progress occurs. Outcomes that either entail all, some, or no incumbentssurviving the technological jump are all supported in the model’s different regimes. The model,however, focuses on when entrepreneurs emerge regardless of whether incumbents survive ornot.Entrepreneurship is worthwhile studying because it is so important in advanced economies whereit is attributed with a large share of employment and a very significant share of radicalinnovations. Among this essay’s recommendations to promote entrepreneurship is a call forcompetition policy be expanded to deter collusion on technology policy by industry incumbents.The second essay, “An Auction Solution to the Joint Venture Prisoners’ Dilemma”, recommendscertain methods for improving economic efficiency when firms do collude in technology giventhat the collusion is in the form of a joint venture.Joint ventures, especially of the R&D type, are becoming increasingly important as a way togain needed technological and market competencies. Unfortunately, many joint ventures havethe characteristics of a Prisoners’ Dilemma. Firms may cooperate or defect in the venture. Ifcontracts, side-payments, and third-party verification of the venture outcome are unavailable,2then the dominant solution to the Prisoners’ Dilemma (mutual defection) results. This paperproposes the use of an ex-ante auction to obtain a Pareto-improvement for these ventures. APareto-improvement is assured when non-transferable costs and benefits of firms are notstrategic. When this condition is not met, certain restrictions must be met in order to obtain theefficiency improvement.The auction effectively creates a competition for ownership of the joint venture that allowsredistribution of the payoffs to the firms through the bids so that the firms eventually will shareequally in the sum of the venture’s outcome. Under such a transformation (which makes thegame dynamic) each firm’s optimal strategy becomes maximizing the total payoff, and thiscorresponds to the jointly efficient outcome.Although other economic papers have used auctions and buy-out options to improve efficiencyunder various non-cooperative circumstances, the mechanism and application presented here isunique. Furthermore, the straightforward implementation outlined can be readily exploited bypolicy-makers in appropriate cases.The third essay, “Trade with an International Open Access Resource and ImpeifectCompetition”, also contains some strategic options that policy-makers may find attractive forimproving the welfare of their country, in this case, under trade.This paper analyzes the problem of trade between countries that share an international open3access resource. Addressing this problem may be increasingly important as the world reachesthe limits of critical shared resource stocks. The problem is modelled as a one primary factor,two intermediate goods, one final good (that is harvested from the open access resource), twonation world. In this world, it is assumed that either the trading game between nations takesplace over one stage so that nations are price-takers, or that it takes place over two stages so thatthe nations take advantage of their market power. Imperfect competition and the externalitiesarising from the existence of the open access final good stocks affect the trading efficiency, andtherefore the incentives to trade.For example, analysis shows that no trade will occur when the open access resource stocks arerestricted to a level below the combined autarkic harvesting capacity of the nations involved.Analysis also shows that, under certain conditions, one nation will favour the two-stageproduction and trade structure although joint welfare would be improved under the one-stagegame structure.This paper is meant to complement the trade literature, specifically the literature on strategictrade and the literature on trade of shared limited resources. While it draws from these sources,it also introduces different modelling assumptions which provide some different policyrecommendations. The policy areas considered include trade, competition, and R&D funding.For example, this essay recommends that R&D funding be directed to areas where a nation hascomparative advantage in order to improve that nation’s (and the world’s) welfare.4While these three essays do differ in their focus, they all provide an analysis that differs fromexisting literature in the area. The essays also generate some useful policy recommendations toconsider. They show that formal analysis of common economic phenomena, when approachedfrom different perspectives, yields valuable insights that are worth considering by both publicand private policy-makers.5CHAPTER ONE: ESSAY ONETECHNOLOGICAL FORCE: THE EMERGENCE OF ENTREPRENEURSHIP1. IntroductionThe emergence of entrepreneurship is a continuous and important phenomenon in moderneconomies. In retrospect, it is often entrepreneurs that exploit opportunities available to anincumbent. In most classical economics literature there is no room for entrepreneurs becauseincumbents do not make mistakes, and because a static game is often assumed. Where the newerliterature provides room for the entrepreneur, it is often because there are exogenous differencesbetween entrepreneurs and incumbents, or because incumbents do make mistakes over time. Thisessay takes the view that there can be opportunities for the entrepreneur even when incumbentsare perfectly rational, have perfect foresight to future technologies and have access to perfectcapital markets, but are precluded from coordination amongst themselves.So why do entrepreneurs often take advantage of opportunities that incumbents, in retrospect,could have seized? Perhaps the incumbents were focused on the short run competition amongstthemselves instead of on the long run competition amongst themselves and new entrants.Consider, for example, the opportunity that arose when vacuum tubes were replaced bytransistors in the 1950s. All the top incumbents failed to take advantage of this event, and it isnewer firms that are at the top of the semiconductor industry today. Foster (1986) also citesother examples that show incumbents waging the competitive battle in the short run to their6detriment in the long run. Incumbents, in order to survive in the present, devote resources toachieve high current efficiency instead of using them to exploit more attractive opportunities inthe future. When this occurs, future entrepreneurs whose resources are devoted to these newopportunities will have a competitive advantage over these incumbents. This essay presents amodel of how such opportunities for new entry can occur when incumbents are caught in aPrisoners’ Dilemma game.Most of the literature analyzing the failure of incumbents to exploit a new opportunity focuseson differential incentives and on structural inertia. It is often the case that incumbents lose apatent race because they have different incentives than an entrant or follower. Reinganum (1985,1989) and Beath et al (1989) provide examples of where an incumbent chooses to allow a newentrant to exploit such a foreseeable opportunity because the incumbent has less of an incentiveto battle for that opportunity than the entrant. Even on in international context, Brezis et al(1993) show that if a new technology is less productive in the short-run but more productive inthe long-run than an existing technology then only the follower nation will adopt it. Foster(1986), Henderson and Clark (1990), Morison (1966), and Hannan and Freeman (1984) allconsider boundedly rational incumbent firms that fail to take advantage of exploitableopportunities because of an inertia which creates myopia, added cost, and delays which theentrants do not experience.It appears that the consequences of being an incumbent are understood from the perspectives ofincentive effects and inertia effects. However, the literature has not explored as fully the choices7of incumbents which take into account these adverse consequences. Consider one choice that anincumbent may have - it can structure itself to achieve a high efficiency in the short-run at a costof high inertia or it can structure itself to achieve a higher efficiency in the long-run but at a lowefficiency in the short-run. If that incumbent is not under pressure to achieve high efficiency inthe short-run, it may choose to think long-term. However, in many circumstances, due to eitheror both competitive pressures and stakeholder pressure on the firm to be profitable in the short-term, such an incumbent is usually required to achieve short-term efficiency. As a result, mostof the time, the consequential inertia and incentive effects that are described in the literature areobserved.This essay differs from this literature in that it explores the decision given the consequences. Itis also different in that it does not give up the assumption of perfect rationality and foresight ofthe firms that is noted in some inertia literature (like Henderson & Clark (1990)); nor does itgive up the certainty assumption (i.e., the ability to exploit a new opportunity is not stochastic)that is found in some patent race and leapfrog literature (like Reinganum (1989) and Beath etal (1989) respectively). Instead, this essay uses the inertia, leapfrog, switching cost, technologychoice and entry game literature as a basis to compute the payoffs and consequences of thestrategic game of choosing which temporal focus to pursue in a competitive environment. Thechoice of temporal focus is embodied in a choice of technology strategy - the choice betweenbeing statically or dynamically efficient. Where static efficiency is defined as being efficient inthe short term, dynamic efficiency is defined as being even more efficient in the future butrelatively inefficient in the short term.8Many authors, including Schumpeter (1934) and Klein (1977), have pointed out that it isunusual, if not impossible, to be simultaneously efficient in a static and dynamic context in acompetitive, changing environment. Trade-offs must be made in reality, as in this model. Thecompetitive environment in this model can encourage specialization and “lock-in” by firmsseeking high static efficiencies which may be to their long-term detriment; or, it may encourage“flexibility” by firms seeking high dynamic efficiencies which may turn out to be to their short-term detriment. With such competition present, added pressure is put on the incumbentschoosing lock-in or flexible strategies to the point where they may choose a non-optimal strategysimply as a reaction to what they feel their competition may choose. They may fmd themselvesin a Prisoners’ Dilemma, where individual incentives lead to a jointly sub-optimal outcome forthe firms.After such a sub-optimal outcome occurs, observing firms and other potential entrants may thentake advantage of the situation. In the model presented below, entrants with the new technology -the entrepreneurs - emerge as a result of the Prisoners’ Dilemma outcome occurring betweenthe incumbent firms. Therefore, under the Prisoners’ Dilemma game, incumbents who wouldlike to and could exploit future opportunities for higher profits do not as it is assumed that theycannot collude amongst themselves. Instead, they rationally choose to “focus on short-term”competition leaving the entrepreneurs to exploit the future opportunities and eventually forcethese incumbents out of the industry. This choice of focus is the rational choice that ensuresthem the highest individual profits over the game when they cannot collude.9The game will be defined and then examined in the remainder of this essay. The game’soutcome will be studied and an important extension considered. Analysis of the model willprovide information as to how incumbents are forced out of the industry, how entrepreneursemerge, and whether this emergence is beneficial. Section Two provides definitions of the game,an analysis of a specific example of the model, and an important extension. Section Threecontains the important conclusions.2. The Model2.1 Important Definitions:The emergence of entrepreneurship in this model is largely based on the technology strategychoice. For simplicity, each firm has a choice of playing one of two technology strategies in itsfirst period of existence in the market. The two choices are “lock-in” and “flexible”. Thesecorrespond roughly to choosing between static and dynamic efficiency. If a firm locks in, it canproduce at a lower cost than non-locked-in competitors for its period of lock in, but it is“impaired” in its transition to a new, even-lower-cost production technology in the futurecompared to such a transition by non-locked-in firms and new entrants. If a firm stays flexible,it has a higher cost of production than locked-in firms for the periods it chooses to stay flexible,but it suffers no impairment to switching to new technologies when those technologies emerge.It can switch to a new technology at the same cost as a new entrant can learn that technology10(assumed a zero cost here), whereas a locked-in firm incurs some time delay or some positiveadditional variable cost to make the change’. Furthermore, staying flexible guarantees thatincumbent a “spot” in the second period as it is assumed that by its experience of flexibility inperiod one that it has an even lower variable cost than new entrants in the second period, butonly lower by some epsilon very near zero.In this model, it can be assumed that locked-in firms either require one period to adjust or incurhigher costs to adjust. The period of adjustment is one of going through the process of“unlearning” the locked-in process. While this adjustment is taking place, it could be assumedthat the firm stays at its production efficiency level or that it exits for a period if need be (in theexample, it does not matter as the locked-in firm is driven out regardless).The higher costs to adjust are incurred because it is assumed the locked-in firms must pay apositive price to “unlearn” the locked-in skills. The cost to “unlearn” as this essay describes theprocess, is similar to the disposal costs incurred with highly specific capital that is no longer thebest technology. In this essay, capital is not necessarily the only area in which a firm incurs the“disposal” or “unlearning” cost. Other areas could be in changing processes, technologies,1 Consider the technology strategy choice as a choice between an investment in either basic or specific R&D.Cohen and Levinthal (1990) show that a firm may lack knowledge diversity if it chooses to invest in specificR&D. This lack translates into a low “absorptive capacity” - a measure of a firm’s ability to recognize,assimilate, and apply new external information. With low absorptive capacity, it is reasonable to assume thata firm will incur some time delay or cost to switch to a new technology.11organizational designs, corporate philosophies, or manufacturing structures2.Now, it must be noted that the adjustment process in no way takes away from the firm beingfully rational. Being fully rational allows a firm to be able to costlessly compute any strategychoice and its payoffs and other implications. It does not ensure that the firm can produce at anytechnique that comes along, or even be able to “think” differently in investment terms than itdoes in its locked-in strategy. For example, the locked-in firm is unable to efficiently invest in,or otherwise finance, another plant producing with the new technology because that locked-infirm does not know how to think in terms of that new technology and therefore cannot efficientlyconstruct or monitor its investment. All that the locked-in firms know is that the new technologyis better and by how much, but they cannot instantaneously use it.The potential entrants and the flexible firms do not have to go through the unlearning processand can take advantage of new technologies without delay. This is how new entrants with newtechnology- entrepreneurs - can emerge.2 Other support for the intuition that inflexible strategies are worse than flexible ones in the long-term in adynamic environment are found in the areas of: retooling costs, FMS, scarce resources, the sunk-cost “fallacy”,and some organizational behaviour effects. Retooling can have higher costs than tooling-up because of: firingcosts, retraining or retirement costs (see Bartel & Sicherman (1993)), and costs of disposing or maintaining old-technology facilities. Flexible manufacturing systems (FMS) have shown in many circumstances (both real andtheoretical) to be less costly in the long run than dedicated machines when the production environment isdynamic (see Roller & Tombalc (1990)). Although perfectly rational actors may not be susceptible to the sunk-cost fallacy, real people often increase the costs of now-inefficient locked-in technology by continuing to useit (possibly due to some reputation considerations) instead of switching to better technologies. Any change intechnique usually incurs some organizational change. Due to some institutional inertia and personal inertia ofemployees, there may be extra costs to change from a well-established condition over that from a flexible (ornew) condition (see Rumelt (1994)).122.2 Definition of the Game:In its simplest form, the model is a two period game involving Cournot competition. There aretwo incumbents in the market. At time t=O, these incumbents must choose their technologystrategies, and then play Cournot output production strategies against each other. At time t= 1,similar decisions must be made by the remaining incumbents and any new entrants. The gameends at time t=2. Profits can be made in period one, between time t=O and t== 1, and in periodtwo, between time t= 1 and t=2 (see Figure One for timing). Assume that at t=O a technologyis available in either its flexible or lock-in forms. At t= 1 a new technology is available in eitherits flexible or lock-in forms. The occurrence of the new technology was foreseen by firms att=O and t= 1. This new technology is superior to the first technology in either form (i.e., thenew technology’s lock-in form allows a lower variable cost than the old technology’s lock-inform).Assume that the incumbents are symmetric to each other, and that the entrants are symmetricto each other. Entry is blocked at t=O (for all of period one) where only two incumbents areallowed. Entry is free at t= 1 (for all of period two). Exit is considered free throughout thegame.Without loss of generality, this model focuses on the simplest form of competition - a duopoly.The relevant player space is composed of the incumbents {Il, 12}. This occurs because the newentrants have their strategies and population completely defmed by the model’s assumptions and13The Timing of the Simple Game:Figure One: Time Line of the Modelthe play of the incumbents. The number of entrants is calculated from the inverse demandfunction and the production cost functions of the incumbents assuming Coumot competition. Thetechnology strategy of the entrants will be the dominant strategy of locking-in because when theyenter it is the last period in the game and there is no point to being flexible then. Thus, theentrants’ strategies are completely defined by the model’s assumptions and the entrants’population is completely defined by the inverse demand function and technology strategy choiceof the incumbents.t=operiod onet=1period twot=2Start of GameTwo incumbents, whodecide their technologystrategies and resultmgproduction levels, existat t0.Only Technology Cl isavailable to them, ineither the lock-in orthe flexible forms.End of GameAt t1 incumbents decide whether to exit.Entrants decide whether to enter. Then allremaining firms decide their own technologystrategy and their resulting productionoutput level under Cournot competition.Technology C2 is now available to them,in either the lock-in or the flexible forms.This technology is superior to the old one(j.e., C2’s lock-in variable cost is lower thanCl’s lock-in variable cost).The occurrence of C2 was foreseeable at t0.14Production output strategies are assumed Coumot (i.e., incumbents choose outputs knowing themarket determines the price to clear the output). As the entrants’ actions and the incumbents’output choices are fully defined, the only strategy space to consider is that of the technologystrategy choice {flexible, lock-in} of each incumbent over each time period {period one, periodtwo}. It is relatively straightforward to compute the payoffs of the two incumbents over the twoperiods for each technology strategy in order to analyze the game in a 2x2 (number ofincumbents x number of technology strategies) normal form (see Figure Two).While the equilibrium in production output is assumed to be Cournot, the equilibrium in thetechnology strategy game is assumed to be the dominant strategy solution to the normal formgame where choices are made simultaneously in each period by the relevant firms.When the incumbents are assumed to be symmetric there are only four different normal formpayoffs to consider: c, d, w & s3.The normal form could take one of many well known structures such as a Prisoners’ Dilemmaor a Co-ordination game. When the normal form is a Prisoners’ Dilemma game each incumbentis “forced” to lock-in due to the presence of its rival although it would rather pursue flexibilityAsymmetric incumbents complicate the analysis by adding four more payoffs to consider. Exploring theramifications of this complication is straightforward and a possible future extension to this game.15c,c s,ww,s d,dFigure Two: The Normal Form of the Modelif it could collude with the other incumbent on strategy4.When the competitive pressures forceincumbents who are capable of blocking the emergence and want to block it to rationally focuson the short-term, future opportunities are exploited by entrepreneurs instead.Only lowest-cost firms can stay in the game in the final period as all others incur a loss. Thus,non-credible threats made by non-minimum-cost incumbents to stay in the market are ignoredby potential entrants. The reason that only lowest-cost firms can stay in the game in the finalThe analysis is more complicated when analyzing the model if the incumbents could also collude on output aswell as on technology strategy choice. Efficiency would dictate that the strongest competition available (i.e.,static efficiency and Cournot outputs) would lead to the highest welfare. The results of this paper are consistentwith such a proposition.The Normal Form of the Simple Model:(two period game)12F LIiFLRead Cells as:payoff Ii, payoff 12Ii: Incumbent One12: Incumbent TwoF: flexibile technology strategyL: Lock-in technology strategyProfits summed over both periodsProfit for a period rev - costs of periodFirst period profits are Cournot duopoly16period is that free entry forces out any inefficient producers. Consider the final “space” opento a firm to profitably exist in the market where the number of such spaces is calculated fromthe Cournot equilibrium in firm outputs. Imagine that either an efficient (lowest variable cost)firm or an inefficient (higher variable cost) fmn is considering existing in that space. If eithercould exist and make a profit, it is logical to assume that the more efficient firm would win thespace. The mechanism that would enable this to happen is a War of Attrition game wagedbetween the two firms vying for that space taking place over a number of “sub-periods” thatmake up the period in question. Thus, split period two into a number of subperiods and play theWar of Attrition game. Either firm is willing to stay in the market through period two’ssubperiods until its expected profits equal that of not being in the market in the first place. Sincean efficient firm’s profits are higher and losses are lower per subperiod, it will stay in longerthan an inefficient firm. The inefficient firm knows this, so it exits at the beginning for a zeropayoff in the second period instead of incurring a loss. The efficient firm knows this, so itenters. Note that at all times it is assumed that the firms continue to play Coumot outputs basedon the number of firms in the market.Assuming that this model contains a War of Attrition Game over its second period leads to thefollowing proposition.17Proposition One: If an incumbent locks in the first period, regardless of the choice of theother incumbent, that locked-in incumbent will be forced out of the gamein the second period.Proof: Consider the locked-in incumbent who, in the second period, has a highervariable cost than any entrant or any flexible incumbent. Therefore, all lower costfirms that can enter will do so until there is a game between the final entrant whomay enter profitably and the locked-in incumbent who wants to stay in themarket. Consider this encounter in the following:i) It may be the case that parameters are such that an “extra” entrant canenter and make a profit under Coumot competition whereas the incumbentmakes a loss. In this case it is straightforward that the incumbent mustexit and allow the entrant to displace it.ii) If it is not the case above, the only other case to consider is the lessstraightforward one when there is a coordination game. If both the locked-in incumbent and the lower-cost firm (the entrant) both stay out of themarket then they both make zero profit, if one stays out and the other isin then the outsider makes zero profit and the producer makes a positiveprofit (higher for the lower-cost firm), and if both produce under Cournotcompetition then each makes a loss (larger for the incumbent). So what is18the outcome5? Consider the War of Attrition game described aboveoccurring within period two between an efficient firm, the entrant, and theinefficient firm, the locked-in incumbent. In this case, the incumbentrationally chooses to stay out of the market in the second period.Therefore, in either case, no locked-in incumbent exists in the second period.DIt can be concluded from the preceding proposition that any incumbent who chooses to bestatically efficient in any but the last period will be forced out of the market in the subsequentperiod. This explains how incumbents, who are capable of blocking entry and remaining in themarket, can be forced out of the market by new entrants when those incumbents choose to bestatically efficient at some time. In order to explore the remaining questions of howentrepreneurs emerge and whether that emergence is beneficial, a specific production-costfunction and demand function are assumed. However, before answering those questions, it isimportant to clear up a few points regarding the play of incumbents in the general gamedescribed above.There are other possibilities to consider if the War of Attrition is not assumed. One possibility is that a MixedStrategy Equilibrium is “non-sensible” as it may result in neither firm entering a profitable market (and eachfirm making an expected zero profit in the equilibrium). An other possibility is that an ordering scheme for entryinto period two could be used that is based on expected profits and experience, but it has no theoretical basis.19The points of clarification have to do with some details of the game that are not relevant to theequilibrium paths of interest. The points deal with what strategies are implemented by firmswhen they incur a negative profit in some period. The first point to note is that this modelassumes that if a firm’s profit from competing in the second (and final) period is negative thenthat firm does not enter the market in that period and thus records a zero profit for that period.The second point to note is that the model assumes that if a firm’s profit in the first period isnegative but its profit in the second period is positive and large enough to make the sum overthe two periods positive then that firm produces in both periods. The reason for this strategy isthat it is assumed that an incumbent is guaranteed a “spot” in the second period only if it isflexible in the first period (i.e., it is assumed that by its experience of flexibility in period onethat it has an even lower variable cost than new entrants in the second period, but only lowerby some epsilon very near zero). Thus, if that incumbent chose to stay out in the first periodinstead of taking the loss it would not be guaranteed a spot in the second period as there is freeentry in that period and it would be on par with the “large” number of entrants vying for spotsin the market at that time.With these details cleared up, the questions of how entrepreneurs emerge and whether thatphenomenon is beneficial can now be answered.202.3 Analysis of a Specific Game:With some loss in generality6,assume a linear demand specification:P=A-BQ, p=l,2where subscript p is the period, P is the market price, A is the price intercept set at some valuegreater than any of the costs for both periods, B is the demand slope, and Q is the totalproduction of the firms present in that time period. For simplicity, assume that A and B remainconstant over the two time periods7.Assume that the cost function for firms in the industry is made up of two components: a variablecost and a fixed cost. The form of the variable cost is simply a constant marginal cost multipliedby the quantity produced by the firm. The fixed cost is a constant amount incurred by any firmwhich produces in a period8.The profit function for firm i derived from the costs assumed is:fl = (F,, - c) q14,- f,, , p = 1, 26 The use of a general demand specification, and the use of alternative demand specifications are possible futureextensions to this model.‘ Considering A and B parameters that change over time is a possible future extension to this model.Considering fixed costs that are asymmetrical or that change over time is a possible future extension to thismodel.21where: f is a fixed cost of production incurred in each time period the firm is producing; c isthe constant marginal cost of production in period p; and q1, is the firm’s output in period p.A positive fixed cost of production is assumed in order to obtain a fmite Cournot equilibriumpopulation. For simplicity, assume that fixed costs remain constant over time and that allproducing firms incur the same fixed costs regardless of variable cost.The Cournot reaction function for each producing-firm involved is derived from the assumeddemand and cost structure as:q* = — B)Dci — C1p; j.,p = 1, 2Assume that the parameter restriction c,, < C1 < Cf < A holds, where ç is the lock-in variablecost in period two (the variable cost chosen by all new entrants and any firms which chose tobe flexible in the first period), c1 is the lock-in variable cost in period one, and Cf is the flexiblevariable cost in period one. Therefore, an incumbent who chooses “flexibility” will incurvariable costs of Cf m period one and ç in period two, while an incumbent who chooses “lock-in” will incur a variable cost of c1 in both periods9.Firms will exit when it is rational to do so. In the last period of the game if a firm sees that itwill incur a loss it will exit. However, if it incurs a loss in the first period and can calculate acompensating gain in future periods, it has the ability to weather the loss and hold a debt in thatAssume that if the incumbent wanted to switch to a new technology in period two that either it must be delayedone period, or incur some cost to do so. Here, for simplicity, assume that the cost equals the difference betweenc,, and c1. The cost only needs to be positive, in general, for the results to hold.22first period in order to assure itself a position in the future period where it profits.As assumed, the beginning of period two, at t= 1, a new technology emerges that is superior tothat used in period one. That emergence was foreseeable to each incumbent at the beginning ofthe game. Given that this second period is the last one in the game, the only rational choice inperiod two for technology strategy is to lock-in as there are no benefits from staying flexible inthe last period of the game. Therefore, all flexible firms from period one and any new entrantsinto period two play the lock-in strategy in period two. Lastly, but most importantly, assumelocked-in firms are impaired in their ability to make the transition to a new technology.Under the general model assumptions and given the linear inverse demand specification, theprofit specification and the cost restrictions the normal form can be computed for actualallowable parameter values. It can be seen that the possibility that a Prisoners’ Dilemmastructure will never be the result under all parameter values is zero. Appendix One provides thealgebra under the assumptions outlined. It is straightforward to show either by numericalexample (see Appendix Two) or by analyzing the restrictions that some allowed parameter valuesprovide a Prisoners’ Dilemma structure in the model’s normal form (see Appendix Six).Figure Three presents the normal form of the game where a Prisoners’ Dilemma arises. Itrequires that the payoffs are ordered such that symmetric flexibility dominates symmetric lockin, but that the individually dominant strategy is lock-in when considering either a lock-in orflexible rival. A further payoff restriction is often also included as part of the definition of a23c,c s,ww,s d,dFigure Three: Prisoners’ Dilenuna Payoff Ordering RequirementsPrisoners’ Dilemma - that symmetric flexibility also dominates the swapping exploitation of arival’s flexibility (i.e., playing (F,L), (L,F), (F,L) ,...). This further restriction is not appliedin this model. One reason is that this model analyses only the one-shot Prisoners’ Dilemma sothe swapping is not an implementable option of firms or policy-makers. Another reason is thatas soon as a firm locks-in it is out of the market in the next period and not available to returnany swap to its partner. Lastly, this further restriction does not alter any of the conclusions orpolicy implications of this essay. Therefore, any mention of the Prisoners’ Dilemma structurefollowing will not assume that this further restriction need apply.Payoff Ordering Required to Get Prisoners’ Dilemma Game(two period game)12F LFIiLFor PD Normal Form, the payoffs must be ordered as follows:w>c>d>s where d>Oalsoassumed.The extra condition that c + c > w + s is not required here.24The result that a Prisoners’ Dilemma structure can arise is significant. It means that, undercertain regimes, profit maximizing incumbents who would choose to be flexible if they couldcollude with one another choose instead to lock-in because they must compete in the short run.As a result, new entrants with new technology (i.e., entrepreneurs) are allowed to exploit theopportunity in the future that the incumbents cannot, but would have liked to.Regardless of whether the normal form produces the Prisoners’ Dilemma game or not, entry bynew firms occurs in the final period whenever an incumbent locks-in in the first period. Holdingall else constant, the reduction in variable cost embodied in the new technology allows at leasttwo low-cost firms (composed of incumbents or entrants) to produce in the final period. If oneincumbent is no longer low cost, then entry must occur. The following proposition results.25Proposition Two: There is entry in period two whenever an incumbent locks in, with moreentry occurring as more incumbents lock-in in the first period. There maybe entry in the second period even if all incumbents are flexible in thefirst period.Proof: As shown in Proposition One locked-in incumbents with higher variable coststhan entrants in the second period get forced out. Thus, if one incumbent locks-in, it must be replaced by at least one entrant in period two. It is straightforwardto reason, then, that if another incumbent also is locked-in it too will be replacedby at least one new entrant. Thus, the more incumbents lock-in the more entrythere will be.To get entry to occur in the second period even if both incumbents remainedflexible requires the following:(A - c)IN7EGER[ fl] 4where the operator returns the integer portion from the contents of the squarebrackets if it is positive, otherwise it returns zero.As none of the parameters are fixed in value at this time, this condition is easilymet. For example, if ç, f, and B were set, an A sufficiently large will satisfy thiscondition and entry will occur even when both incumbents lock-in in the first26period10.DKnowing that entry will occur in each case where at least one incumbent locks-in in the firstperiod the cases where this lock-in occurs can be defined. Lock-in can occur in three cases. Itwill be the dominant strategy for both incumbents to lock-in if the normal form is a case of thePrisoners’ Dilemma game, or if it is the case where the lock-in is both dominant and jointlyefficient for the firms. Lock-in can also occur in a Co-ordination game where a mixed strategyequilibrium entails playing the lock-in strategy with some positive probability. The case wherelock-in will not occur is when the dominant and pareto optimal strategy for the incumbents isto remain flexible. No other cases are possible for the normal form to take, given theassumptions in this model. The parameter restrictions for each of these cases is given inAppendix One. It is straightforward to verify, through numerical example (see Appendix Two)or analytically, that all of these four cases are possible normal forms and that no other non-pathological case is possible (see Appendix Five).Obtaining the result is not an extra-ordinary event. It is already known that:(A - c)IN7EGER[ ] > 3Because for d>0:(A-c)IN7EGER[ 1] 3and c < c1. Given these conditions, it is clear that the parameter values that allow entry into the second periodeven when both incumbents are flexible is a large subset of the parameter values that meet the necessaryconditions that d >0 and c, < c1.27These four cases cover the three possible outcomes that can result in an industry when atechnological advance occurs. All incumbents can remain in the industry after the advance, aspredicted by the flexible dominant normal form. Some but not all incumbents can remain in theindustry after the advance, as predicted by the Co-ordination game normal form. Or, noincumbents can survive the advance, as predicted by the lock-in dominant and Prisoners’Dilemma normal form. Thus, the model of the effect of a technological advance on industrystructure presented in this essay is robust to all possible observed outcomes.Consider now the welfare implications of entry. When entry does occur, it may be because atleast one incumbent locks-in in the first period. When this occurs at least one firm in each periodis statically efficient- uses the lowest variable cost technology available. When firms arestatically efficient they may end up making increased profits, but it is a certainty that the priceand quantity to the consumer is improved given Cournot competition. It may then be that totalfirm profits, consumer surplus, and total welfare are all increased by the occurrence of entrywhen there is lock-in by at least one incumbent. Thus, the proposition follows.28Proposition Three: The profits summed over all producing companies (Total CompanyProfits), Consumer Surplus, and Total Welfare are all increased whenentrepreneurs emerge (i.e., when both incumbents lock-in).11Proof: i) The only difference in the outcome of the full game between the casewhere both incumbents lock-in and where both incumbents chooseflexibility is in period one. Therefore, only the change in profits inperiod one will be analyzed (i.e., the difference in profits between thecases of (L,L) and (F,F) under Cournot output):A LL-FF = (——) [(A — c,)2 — (A — C?2] > 0as A > Cf > C1.ii) Similar to the argument above, look only at the period one differencebetween (L,L) and (F,F) strategies regarding consumer surplus (the otheradditions to consumer surplus between these conditions cancel out):ACSIJFF = (-h) [(A - c,)2 - (A - C)] > 0as A > Cf > c1, where:“ As the incumbents here are assumed to have symmetrical payoffs, only the cases where they pursue the samepure strategies will be analyzed here. The extension to the mixed strategy equilibrium in the Co-ordination gameis a simple case (see footnote in Appendix Five).29Q=Eq2iii) The total change in welfare, then, is the sum of the two elements above:AWELFAREIJ.FF = (—-_) [(A - c1)2 - (A - c)2] > 0as A > Cf > C1.This welfare measure does not take into account of the extra costsinvolved due to the emergence of entrepreneurship: the cost of twobankrupt incumbents plus the cost of setting up two replacing entrants.Thus, if the change in the welfare is larger than the costs of having newentrants emerge then the total net change in welfare is positive. Therefore,policies that ensure the normal Prisoners’ Dilemma outcome occurs areworth pursuing’2.It can be concluded that entrepreneurial entry (which occurs when both incumbents lock-in inthe first period) is welfare increasing under certain restrictions. What drives this result is the12 Consider a numerica’ example which shows the welfare benefits. If c=4,c1—3.5, A=9.O, B=1, f=3, thenthe gross change in total welfare over both periods = 2.38, and the total welfare generated by collusivetechnology strategies in period one = 5.56. Therefore, the percentage change in total welfare with respect tothe total welfare generated by collusive technology strategies in period one is a significant 42.8%.30presence of statically efficient firms in each period that the industry exists. When an industrycontains only lowest-cost firms in each period of its existence, that industry can be defined asbeing “dynamically efficient”. Such an industry can occur in this model. Of the cases where thisdoes occur, perhaps the Prisoners’ Dilemma normal form of the model is most interesting. Here,firms lock-in due to competitive pressures resulting in incumbents being statically efficient inthe first period and locked-in entrepreneurs who fully occupy the market in the last period beingstatically efficient in that last period. The question of how entrepreneurs can emerge even whenincumbents are capable of blocking them is thus answered in this model’s Prisoners’ Dilemmaoutcome. Further, the question of whether that phenomena is welfare improving is also thusanswered.If bankruptcy (exit) and entry costs are low, and the number of firms incurring them is low,social efficiency can improve as a result of this type of competition where entrepreneurs emerge.It makes sense that it is “socially beneficial” that some firms choose static efficiency and thenare eliminated by future statically efficient firms in a path to step-by-step dynamic efficiency forthe industry (and society) overall. Therefore, by having firms that are statically efficient at eachperiod in time, dynamic efficiency is assured in the industry. The machinery behind thisefficiency is competition and entrepreneurial entry.Given that such market evolution is beneficial, there are obvious policy implications to be takenfrom this model. Some of the assumptions of this model can be supported in the real world whenit is beneficial to do so (e.g., when the PD normal form is possible). For example, lower cost31access to technological information may enable the assumption that entrants are up to the latesttechnology to hold true; and some subsidization of firms may enable the assumption of low entryand exit costs to hold true. Such policies could be supported with taxes on the increasedconsumer surplus. Another more direct example of how policy can ensure model assumptionsin the real world is through the implementation of strong Competition Law to ensure theassumption that incumbents cannot collude on technology strategies holds.2.4 Considering an Extension to the Game:One question to consider once this entrepreneur-creating situation is found to exist is just howprevalent is the phenomenon? To answer this question some further definition has to be givento the range (of applicability in the parameter-space) of the model. Once this is done somemeasure of potential entrepreneurial activity based on this model can be computed.The further definition of the model entails putting some further restrictions on its applicabledomain. First, the asymmetry of entry protection between the two time periods must beaddressed. To do this the first period has to be opened up to any potentially profitable entry.Second, restrictions on the model’s variables must be confirmed (they are summed up by theinequalities: c, < c1 < Cf < A). With the range of possible variable values restricted to themodel’s applicable domain, it is possible to set out the restrictions that must be met to achievethe Prisoners’ Dilemma (PD) and the Co-ordination (CG) game situations that may give rise to32what this essay defines as entrepreneurship.These restrictions consist of equations that provide a specific order to the payoffs in the normalform game. The PD situation needs to meet three restrictions whereas the CG needs to meet tworestrictions on the payoff ordering (see Appendix Three for details on all of the restrictions).The “relevance measure” (an estimation of parameter-space area) can now be computed.Unfortunately, the restrictions that identify the game-type are based on Cournot competition withsix degrees of freedom (assuming a linear demand function). Coumot competition entailsdiscontinuous functions (calculating the number of entrants) and non-linearities (calculating firmprofits through reaction functions). Therefore, no closed-form solution to the measure isavailable in even the simple two-incumbent case. Numerical methods can find the measure undera range of variable values.The computer program [listing available upon requesti simply steps by small increments throughthe positive real number line for a number of parameters recording the type of game outcomeresulting from the parameter values. All but one parameter is fixed when the stepping occursfor the one parameter of focus. The game-type is found by evaluating the parameters against themutually-exclusive restrictions defining each of the four game-types. The programming isstraightforward working with linear inverse-demand functions and Cournot competition. Fromthese relevant parameters and the game-type categorization, a number of results can becalculated, including: i) each game-type’s share of the total parameter space; and, ii) the33Consumer Surplus, Firm Profits, and Total Welfare under each game and under each game’spossible outcome (e.g., the change in Welfare between a collusive and a competitive Prisoners’Dilemma type game can be evaluated).The results of numerous computer runs are of some interest. The PD situation occurs inapproximately one-third of the parameter-space in this model (see Figure Eight and Nine inAppendix Four). This is a significant proportion of the parameter-space. It is unknown howrealistic the ranges are for the parameters that give rise to the PD outcome. However,examination of how the parameter space is separated into the differing regimes (of the fourpossible normal forms) reveals a discontinuous and well spread out stratification. Therefore, itmay be safe to assume that, given this spread of parameter values in the space where the PDnormal form occurs, a reasonable proportion of such outcomes will occur in the real world.Therefore, entrepreneurial emergence may arise in a significant proportion of the cases whereincumbents must make the technology strategy decision outlined in this model. It is notsurprising then to observe the phenomenon of incumbents, who are capable of exploiting futureopportunities and would if they could collude with each other, deciding to lock-in and competein the short run and sacrifice the long run opportunities to the new entrants with the newtechnology - the entrepreneurs.343. ConclusionsThis essay has attempted to show that, with slight changes in assumptions, classical economictheory does have room for entrepreneurship. Simply by adding a dynamic element to aclassically-based model of competition and by assuming foreseeable innovations occur as timeprogresses, entrepreneurs do emerge in a substantial range of the model’s area of existence.Further, the process that is created through entrepreneurial emergence can be welfare improving.This is because by having each period’s firms choose to be statically efficient the industryachieves dynamic efficiency over all periods.It would appear that entrepreneurs emerge in the much of the relevant parameter-space of thegames defined in this essay. Entrepreneurs - new entrants with new technology - emerge in thePrisoners’ Dilemma game, in the Co-ordination game, and in the game where the “lock-in”strategy is both dominant and firm-pareto-optimal. No policy action is required to affectentrepreneurial emergence in this latter game. However, policy action can affect entrepreneurialemergence in the Co-ordination (CG) and the Prisoners’ Dilemma (PD) games. In the case ofthe CG, some small subsidy can be given to one or both incumbents to ensure that both do notremain flexible and block entrepreneurial emergence, if that emergence is of net benefit tosociety. In the case of the PD, policies that discourage collusion (ensure competition) betweenthe incumbents (and between all later firms) will help ensure the emergence of welfareimproving entrepreneurs.35Many current competition policies only discourage collusion that has a material effect onindustry competitiveness from a price perspective, while allowing some industry cooperationwith respect to technology-sharing. This essay attempts to argue that competition policy shouldextend to the technology strategy regime as it does to the price (or quantity) fixing regimebecause collusion on technology decisions could be socially damaging. In either case welfare willimprove under most circumstances if larger firms in the market are precluded from colludingunder either regime. In the case of technology strategy this essay shows that when such collusionis restricted overall efficiency will improve. Not only will an industry be statically efficient inevery period but entrepreneurs will have the opportunity to emerge, with all the favourableaspects such entrepreneurial activity has on an economy.Although the conclusions of this essay are based on some strong simplifying assumptions likethe existence of Cournot competition among small oligopolies, fully rational firms, exogenousforeseeable technologies and un-learning costs, it does offer an alternative explanation of whyentrepreneurs emerge in modern economies. Even when it might appear in the real world thatincumbents could have done “better” by focusing on future opportunities and blocking the entryof a lot of the entrepreneurs that ended up destroying them, some incumbents chose not to, andthis essay provides one explanation. It is an alternative explanation to the incentive-differencebased patent-race and technology-adoption literature and to the organizational inertia andboundedly-rational incumbent literature. This essay has presented a model of strategic choice toexplain the emergence of entrepreneurship. Although the scope of the model may be narrowedaccording to the relevant parameter values in the real world, some potentially useful policy36implications can be drawn from its findings.The results presented above are robust to some generalizations of the model. For example, inthe model, the industry lasted for only two periods and the new technological opportunity wasa process improvement. While the extension outlined was important in that it disclosed a“relevance measure” of the model, other extensions are possible and straightforward. It isstraightforward to consider, but less so to do the mathematics, allowing the industry to exist overfurther periods. This extension does not change the conclusions. The conclusions are also notchanged by allowing more than two incumbents in the first period. The model is also applicableto considering the technological improvement as a product enhancement. In this case, firms thatcan exploit this opportunity can offer better product on the market. This translates to better valueto consumers and simply parallels offering a lower price on a set product - a consequence of aprocess improvement.This essay has presented an alternative explanation for why the leadership in tire cords orcomputer equipment or commercial aircraft has switched to entrepreneurial firms. Althoughexplanations may be found in the leapfrogging literature like Brezis et al (1993), in theorganizational structure literature like Cohen and Levinthal (1990), in the technology adoptionliterature like Conrad & Duchatelet (1987), in the technology life-cycle literature like Foster(1986), in the structural inertia literature like Henderson and Clark (1990), in the patent-raceliterature like Reinganum (1989), or in the flexible manufacturing literature like Roller andTombak (1990), this essay provides one more story - a story from a strategic perspective.37CHAPTER TWO: ESSAY TWOAN AUCTION SOLUTION TO ‘TIlE JOINT VENTURE PRISONERS’ DiLEMMA1. IntroductionThis essay contributes to the literature on joint ventures by presenting an implementable methodfor improving the efficiency of some ventures. The essay argues that many joint ventures havethe characteristics of a Prisoners’ Dilemma. An inefficiency arises in such ventures because noparticipating firm has the incentive to carry out the jointly optimal strategy. This essay proposesthe use of an ex-ante auction which makes the division of the venture’s total transferableoutcome uncertain until the auction winner is revealed after the venture ends. When the firmsplay the venture strategies before they know the division, they have an incentive to play thejointly optimal strategy. This incentive makes the outcome of some joint ventures more efficient.According to Taylor (1989) joint ventures are very important in many technology-basedindustries. Roberts and Mizouchi (1989) go one step further in suggesting joint ventures are anecessity in some industries. There is a literature that supports the idea that joint ventures havethe characteristics of a Prisoners’ Dilemma. Parkhe et al (1993) analyze the payoff structure ofinterfirm strategic alliances (ISAs). Their survey of 342 senior executives experienced in ISAssupports the PD payoff structure as a representation of the ISA. Von Hippel (1987) explores38informal trading in know-how between engineers at rival and non-rival firms’. This informaltype of joint venture is also found to be modelled as a Prisoners’ Dilemma.Joint ventures can have the characteristics of a Prisoners’ Dilemma because participating firmscan cooperate or defect on their actions and investments in the venture, often in a undetectableway, until the results become known. In such situations, the firms (and other affected parties likegovernments) would ideally like to solve the PD; that is, to obtain the Pareto-optimal cooperativeoutcome of the venture. There are many ways to solve the PD, depending on what furtherassumptions and circumstances are allowable. With minimal adjustments a straightforwardsolution to the one-shot PD may be available that does not change the competitive nature of thegame and does not require an outside judge to verify which game strategies were played. Byholding an ex-ante auction on the joint venture (JV) outcome, whose winner is not revealed untilthe venture is completed, the PD can be solved. The firms entering the JV will cooperate ontheir actions and each will be rewarded for its cooperation.Non-cooperative game theory provides few, if any, solutions to the one-shot Prisoners’Dilemma. Cooperative game theory provides contractual solutions (which may include side-payments and bargaining). However, there are problems associated with implementing thecooperative solutions. First, it may be expensive to negotiate a contract that is fair to bothparties, and to ensure that the appropriate penalties are enforceable. Second, such cooperationThe know-how trading example provides a simple case of a Prisoners’ Dilemma where the auction solutioncollapses the original game into a trivial optimization. Only the choice of an appropriate bid is important. PDstrategies are irrelevant because the winning bidder will enforce the efficient level of know-how trading betweenthe two firms that it would then own.39might draw the attention of competition authorities worried about collusion. Third, and mostimportantly, the cooperative solution will require some third-party verification of strategiesplayed, which may be expensive or impossible.Finding third parties able to verify what strategies were played by which firms, or even whatthe outcome of an R&D venture was, may be impossible or too costly2, or may be undesirabledue to appropriability problems3. No third-party verification of the actual strategies played isrequired in the ex-ante auction solution presented below. This is because the decision to enterthe auction transforms the game into one in which the division of the total joint venture outcomeis the focus and where knowledge of the strategies played (i.e., how the outcome was generatedex-post) is irrelevant. In effect, the auction redistributes the payoffs to the firms so that eachshares equally in the sum of the JV’s payoffs. Given such a transformation, each firm’s optimalstrategy then becomes maximizing the total payoff, which corresponds to the jointly efficientoutcome.Although this essay may represent the first attempt to use an auction to gain a Paretoimprovement in the outcome of a PD game, there is some history of the use of an auction2 The third party would have to have a comprehensive knowledge of either the companies or the technology orboth in order to judge correctly. If the third party had to verify who defected, for example, by investing badengineers, that third party would have to be able to judge which are good and which are bad engineers in eachfirm, which is a relative measure. If the third party had to verify what the outcome of the venture was, withoutknowing what would have been created had the venture gone on, then that third party would have to haveexpertise in the technology and in the competencies of the venture partners. All of this knowledge is difficultto find in a trustworthy third party. It is furthermore improbable, however, when the firms involved have anincentive to give that third party false information.For example, if the output of the JV cannot be fully protected in some binding way, as through patents, thenthere is a concern that a third party judge would have the opportunity to appropriate some of the output.40towards similar ends in coordination games. Van Huyck, Battallo and Beil (1993) describe howan auction was used to filter entry into a coordination game. That game involved individualsapplying effort in a project, and had payoffs based on differences from the mean effort put forthby all participants. Players could support beliefs (and choice) of higher effort inputs by usingforward induction based upon the entry bid information. In effect, the entry auction allowed theselection of a more efficient Nash Equilibrium by allowing effective communication ofinformation about the problem of equilibrium selection.Beggs (1989) provides another example in which an auction can resolve an inefficiency. Thatauction allows the most efficient firm to signal the government to grant it the rights to a marketthat may have, in the absence of the auction, gone to an inefficient firm4.One alternative solution to the one-shot Prisoners’ Dilemma in the literature that is worth notingis the Lindahi process. Lindahl (1919) showed how an auction process might be used by agovernment to set the level of public spending. This process requires a set of auctions ratherthan one, and cannot guarantee an efficient outcome in general (as shown in Inman (1987)). Infact, Malinvaud (1971) described the many inadequacies of the process including the result thatthe outcome of the process is simply another PD, but one created by government rather than bythe market.‘ Beggs presents a two-period model where the existence of consumer switching costs and firm start-up costs maylead to an inefficient firm winning the market. A government that holds a Vickrey auction for licenses toproduce in the market will be able to ascertain and choose the most efficient firm to produce in that market.41Another solution to the one-shot Prisoners’ Dilemma that is closer to those proposed here ispresented by Frohlich (1992) and based on impartial reasoning. Implementing the “You cut thecake and I choose rule” by having the players in the PD choose strategies without knowingwhether they are choosing for themselves or for the other player allows mutual cooperation tooccur in many cases. Frohlich explains this implementation in theory but does not explain it inpractice where it is difficult to make real players in real situations uncertain about who they willbecome in the future (i.e., the “row” player or the “column” player). Frohlich’s solution forcessome difficult constraints on the players: by forcing the players to be able to switch positions,only symmetric players are allowed who have no personal control over changing the strategyimposed on them whether they chose that strategy or not5. The results of Frohlich’s solution andtype of uncertainty used are similar to those of one of the solutions presented below, but thatis where the similarity ends.Still other solutions to the Prisoners’ Dilemma have been described for the long or infmitehorizon case. Axeirod (1984) writes in detail about various strategies that can be employed inthe repeated PD when playing against various rival strategies (in an unknown horizon game).Guttman (1992) shows how cooperation can emerge in PD-like situations when rational playersact to preserve valuable reputations over time. Similarly, Hakanson (1993) studies structural andFor example, if player i chose to defect while player j chose to cooperate and then the uncertainty about whomust implement whose strategy was resolved such that i had to implement j’s strategy, it must be the case inFrohlich’s solution that i must have control over changing that implementation (without being caught). It isnot difficult to imagine instances in which this is an unreasonable assumption. The solutions presented in thisessay rely on uncertainty about how payoffs are to be divided, not about who will be which player. Frohlichalso fails to mention how to resolve the PD with two Nash equilibria that arises when P> (S + 1)12 (note: hisnotation used to describe the payoffs where P is the payoff from mutual defection, S from being the cooperatoron the other’s defection, and T being the defector on the other’s cooperation).42partner selection issues regarding collaborative R&D ventures. Hakanson finds that repeatedventures where partners can build reputations can result in cooperative outcomes. Kreps et al(1982) study how rational cooperation can arise in the finitely-repeated Prisoners’ Dilemma whenthere is incomplete information on rivals’ types and strategies. Some genetic algorithm work(such as Nachbar (1992) and Linster (1994)) involves finding the evolutionary equilibrium in therepeated PD given an initially random population set where all members of the population areassumed to have limited rationality over the long horizon of the game.In contrast, the auction solution can resolve the PD problem even in one-shot settings with thereis little uncertainty. The focus here is on its application in joint ventures, and the analysisprovides some guidance for business policy. The rest of this essay presents the description ofan application, defines the game, outlines important assumptions, describes the various solutions,details important extensions to the model, and provides some conclusions.2. Description of an ApplicationConsider a one-shot joint venture R&D project between two firms. Assume there is some valuefor the firms in participating in the venture (over using their resources in-house instead of in theventure). For example, complementarities are generated through combining the firms’ equipmentand researchers. Other reasons for joint venturing can be found in the literature. Taylor (1989)writes that R&D JVs are attractive in this globally competitive market because they will43accelerate a firm’s technological development, increase its productivity, and spread the risks ofits technological projects. Roberts and Mizouchi (1989) provide an example of where R&D JVsare not only attractive, but essential: in biotechnology, research collaboration is necessarybecause of the high risks, large capital investments, and multiple technologies required toachieve a successful and profitable result. Ouchi and Bolton (1988) investigate how firms canstructure research efforts to justify large investments that generate leaky (hard-to-appropriate)innovations. One solution they find is the joint venture structure. This structure allows the costof the investment to be spread and the leaky property to be more quickly brought to marketbefore it is imitated by rivals.Given that participation in the JV is beneficial, assume that Firm A and Firm B enter into anR&D joint venture that, if fully implemented, lasts two periods. The output is a valuable newproduct or process. The output is incomplete but of some value at the end of the first period,and of considerably more value when completed at the end of the second period. Both firmsprovide the same input to the venture so that their investment costs are the same at each pointin time during the joint venture6.Property rights are fully defined at each stage of the process. The IV contract is written so thateach participant owns half the IV, and can prove this to a third party. However, what cannotbe proven to a third party is whether the venture is at the end of the first period or the second;6 Assume that either this input is monitored (and enforced) at an acceptable cost or that the input value is non-material compared to the other costs of the firms or the venture. In either case the investment costs axe equalfor each firm at each point in time during the venture.44whether the output is partially or fully complete. It may not be possible, feasible, or attractiveto have an outside judge who has the enough knowledge of the project to ascertain at whichstage the venture is if the venture is supposed to give some hard-to-imitate competitive advantageto its participants.The venture is terminated when at least one of the firms decides to end its participation or whenthe end of period two occurs. Upon termination, the firms can exploit the output from theventure because each of them legally owns rights. Firms are considered fully rational andcompletely informed when they make their simultaneous decisions (i.e., over when to end theventure, and what to bid in the auction).If both firms “cooperate” and work until the end of the second period, the venture is complete.Both firms would then have an equal share-holding of that output and an equal opportunity atthat time to exploit that value on the market. So if the firms have the same opportunities tocreate wealth from the output then each should derive roughly the same welfare from the venture(the net value of which is labelled c).Firms can also “defect” by attempting to exploit the venture prematurely by going to market withthe preliminary product (or to the patent office with the infant idea) without informing the otherfirm which may have been planning to continue the venture. Now, if both firms defect, then theywork only until the end of the first period when the output is not complete. They again split therewards (and the venture ends at this time). The firms would obtain half shares in this lesser45total value of the joint venture (the net value of which is labelled d).If one firm defects while the other cooperates then the defecting firm does well while thecooperating firm suffers. In this case it is assumed that the defecting firm obtains the full valueof the incomplete venture output in the market (the net value of which is labelled w) as it is theonly firm that is prepared to exploit the opportunity at this time. The cooperating firm isrelatively unprepared to exploit the value of the output at the end of period one and it receiveslittle or no positive benefit from the venture at all (the net value of which is labelled s). Theventure ends with the defection at the end of period one so it is assumed that the cooperativefirm cannot continue the research into period two with any hope of future positive retum.If the payoffs order as w > c > d > s with the added condition that c+c > w+s then thegame is a Prisoners’ Dilemma (with all payoffs in these conditions being net payoffs)8.It is also“ This multi-stage Joint Venture game is similar in construction to the so-called centipede game, while having thepayoffs of a PD. In the centipede game, first defined by Rosenthal (1982) and later by Bininore (1987) andothers, each player has a choice of leaving the game or continuing it for future periods. If the players do notleave the game, each player’s payoffs increase in the future. In most centipede games, players take turns inbeing able to leave the game (e.g., the first player has its turn in period one, the second player in period two,etc..). When it is that player’s turn to choose, it can leave with a higher payoff than its rival at that time. If itchooses to continue the game, then the rival can choose to end the game giving the player a smaller payoff thanit could have obtained one period before. However, if the rival chooses to continue the game, the player cannow leave with a higher payoff than two periods before. The Joint Venture game described above does differfrom the centipede game in a number of respects. First, players simultaneously choose to leave (defect) orcontinue (cooperate). Second, the payoffs increase to each player symmetrically and monotonically in eachperiod, provided strategies played are symmetric.8 In the case as described, one possible definition of the payoff is: s=O and w=d+d (with the payoffs beinggross, that is not net the costs of investment when there are such costs).46assumed that c > 0 so that the JVPD is a socially valuable project9.Now consider the payoffs net the investment costs. At the end of each period both firms havemade the same investment. Thus, payoffs w, d, and s all are net the same investment cost’°.Only if both firms cooperate and the venture continues until the end of period two do theinvestment costs increase to both firms11. As these investment costs have no strategicimplications, without loss of generality, assume both of them are zero.With the net payoffs forming a Prisoners’ Dilemma the usual PD outcome would be expectedin this JV. Both firms would defect and only the incomplete product would come to market. Thisimplies a social loss as well as private losses to the participating firms.Now consider the Auction Solution. Here, the two firms place their respective shares ofownership of the venture in a trust that is held until the venture is complete. Then, the two firmsbid for the two shares before the start of the venture. The outcome of the auction only becomesknown when the venture is complete. The auction is of the first price sealed bid variety. Thewinning bidder is given all ownership rights of the venture plus its bid back less a share of theIt may also be the case that d > 0 so that the original joint venture itself is attractive to each firm. In this casethe Auction Solution simply increases the efficiency of the JV. However, when d < 0, then the AuctionSolution allows the JV to occur when normally it would not. Regardless of the value of d, as long as the projectis socially valuable, then paying the costs to implement the Auction Solution may be worthwhile not only to theparticipating firms but also to the government.10 As long as the investment made by all participants is the same (regardless of strategy played up to that periodin time when a new investment must be made) the investment has no strategic implications.“ Thus, for example, payoff c is net i1 +12, the sum of the two investment costs when those costs are assumed tobe non-zero.47total bids while the losing bidder is given its bid back plus the share of the total bids collectedfrom the winning bidder.The auction trust has all ownership rights of the venture until the winning bidder is revealed.When the venture is terminated (i.e., made complete), only afirm which can establish ownershiprights has the legal right to exploit any outputfrom the venture. The ability to claim any payofffrom the venture itself without having won the auction’2 is therefore removed. What the auctiondoes, in effect, is take away ownership certainty, reducing the incentive to defect on another’scooperation. The auction transforms the game into one where defection is no longer thedominant strategy as the distribution of payoffs resulting from any strategy is uncertain until thebids are revealed13.Consider another alternative of writing a contract between firms to form a merger over theventure. The merger agreement would be written so that all profits made by either firm resultingfrom the venture are split evenly. This requires that each firm can be effectively audited by athird party to determine the profits generated by the venture. It is assumed that this is anunreasonable requirement in most cases not only because it would be costly to track and verify12 A firm can still choose to defect under the solution and stop the venture at the end of period one and force theunsealing of the bids. Then, only if it won the auction, and so could prove ownership rights to the venture’soutput, could it exploit the defection by bringing the output to market then. If it lost the auction, the venturewould be over and the other party could bring the output to market as it saw fit. Thus, all previous strategiesare still possible under the Auction Solution.13 The alternative of just having a trust hold the ownerships (without the auction) is pointless. If a trust holds theownerships until the venture ends, a defecting firm can still prove the necessary ownership rights (althoughbeing held by the trust). The original game is unchanged under this alternative, defection remains the expectedaction.48all revenues and expenses of a venture but also because each firm would have an incentive tomisinform the other firm in order to capture more of the profits. The auction solution entails nosuch requirement.The auction creates some uncertainty over which firm will capture the “prize”, and how greatthat prize will be14. As it is structured, the auction provides the firms with incentives tomaximize the size of the prize (by cooperating in the venture), and to maximize their chancesof winning (by bidding up to the efficient levels).When it solves the JVPD, implementing the Auction Solution increases the total rewardsavailable to the firms and, in this example where mutual cooperation corresponds to a betterproduct on the market, also to society.14 The auction creates some uncertainty about allocation of the rewards where the rewards are the shares ofownership of the venture. Bidding on the ownership creates, in effect, a buyout option for each firm. Demskiand Sappington (1991) provide another example of how a buyout option can result in a Pareto improvement.They analyze the use of one in a double moral hazard problem. When there is an independent business entitywhose ownership is transferable, having agents that can be required to purchase it from a principal results inthe agents working harder.493. The GameThe Prisoners’ Dilemma component of the game is standard. Firms have two choices: eithercooperate (C) or defect (D) with the other firm. Firms choose their strategies simultaneouslyknowing the payoff schedule, and having complete information and full rationality. The netpayoff schedule is as shown:Firm BC DFirm C c,c s,wA D w,s d,dwhere: 1. w>c>cl>s2. 2c > (w+s) so that the jointly efficient solution is: C,C3. c > 0 so that firms may participate in the JV under the Auction SolutionAssume that this is a one-shot game. This is consistent with the structure of many joint ventures.Investment in a joint venture will of course be a one-shot game or a finitely repeated game. Ineither case, when there is no uncertainty, perfectly rational firms will defect.Now imagine the following time line. At t=0, which is the beginning of period one, firmschoose their strategies (C or D). At t 1, which is the end of period one and the beginning of50period two, the venture ends if there is any defection. If there is no defection then the venturecontinues until t=2 which is the end of period two. The only times that a firm can leave theventure are at t= 1 and t=2. Assume that no individual opportunity costs or spillovers outsidethe venture exist15. Assume that payoffs meet the Prisoners’ Dilemma structure as defmedabove.If the game stays as described then defecting remains the dominant strategy. Whether the otherfirm cooperates or defects the payoff from defecting is greater than that from cooperating (i.e.,in the case of the other firm cooperating, w > c and in the case of the other firm defecting,d > s).Now, consider the following story: IBM® and Apple® form an R&D JV to create an operatingsystem which will defeat Microsoft® in the marketplace. The JV produces software which thefirms protect by each copyrighting’6.They know that in some months the JV will generate anoperating system capable of matching Microsoft’s. In some more months, they know that the JVwill produce an operating system so good as to make Microsoft’s product unattractive toconsumers. It is unlikely that a third party would be able to ascertain which operating system‘ These individual payoff flows that are outside the joint venture are non-transferable flows - one firm cannottransfer them to another firm through the ownership rights of the joint venture. The only individual opportunitycosts assumed thusfar were the investment costs i1 and 2 which were assumed zero without loss of generality.The reason that these flows are allowable in this game is that they are of no strategic value. The investmentcosts are the same regardless of strategy played at each period in time in the game.It is important that the output produced by the JV have assignable ownership rights. If the JV simply producesideas (which are a public good between the two firms that can be kept secret from third parties), then noownership rights can be transferred and the Auction Solution cannot work. Writing contracts to keep theintegrity of trade secrets would be necessary (but not sufficient) to generate cooperation in such a setting.51is the one representing the full potential of the venture, nor any way to tell who defected if theoperating system is the earlier one. Without seeing the second-period product, a third partycannot make the judgement, and if the venture ends after the first period, the third party neversees the second-period product. Both IBM and Apple may have an incentive to defect on eachother and get out of the JV early. This is because if either cooperates and gets defected on, itis assumed the cooperator will not get to market until well after the defector has made itsoperating system the new standard. This is precisely the kind of case in which the AuctionSolution may provide a Pareto improvement.3.1 The Basic ApplicationNow consider what occurs when the Auction Solution is implemented. At t=O, the firmssurrender their ownership shares to a third party. They then submit bids for ownership of allshares of the venture to the third party. The bids are sealed. The firms then choose and playtheir game strategies (C or D) and the venture is undertaken. When the venture is finished (att= 1 or t=2 depending on the strategies played) the bids are unsealed.The firms do not know the outcome of the auction until the JVPD is fmished - when the payoffresults are attained. Otherwise, they may be inclined to change their strategies. Assume that thisis a first price sealed bid auction. The highest bidder takes full ownership of the prize less somecut of the total amounts bid as its reward. The losing bidder takes the cut from the winner asits compensation. In the case of a tie, a coin is flipped to determine who is the winner and who52is the loser.Assume that to run this auction costs a positive, but very small amount’7,‘y. This cost is splitequally among the participating firms.Formalizing, the ex-ante auction is defined as a first-price sealed-bid where there is no limit seton the amounts bid. The winner of the auction receives the total JVPD payoff plus its bid backless a share, oe (0 o 1), of the total amounts bid. The loser receives its bid back plus thene-share of the total amounts bid transferred from the winner. If there is a tie in the bidding, thena coin-ifip decides the winner’8.It is assumed that the cost to initiate this solution to the PD is non-trivial. This assumption eliminates oneimportant instance of possible multiple equilibria: the equilibrium that may occur when firms do not partakein the auction and end up with the mutual defection outcome, and the equilibrium that may occur when firmsdo partake in the auction and still both defect. Without the y the firms would have the same payoffs under eitherequilibrium. It is not unreasonable to consider that the extra step of partaking in the auction would be costly,so it is assumed to be in all the analysis that follows.‘ There is one further important assumption that is implicit in being able to implement the Auction Solution andthat is the assumption that such a solution can be coordinated between the firms at a non-prohibitive cost.Considering that the two parties had to coordinate to be involved in the joint venture in the first place it is notan unreasonable assumption to extend their coordination a few steps further to allow the terms of the solutionto be set. The agreement on the conditions of the Auction Solution comprises a third-party observable,enforceable contract between them just as the contract that defines the legal entity and ownership of the jointventure does. In either contract, all the terms are settled on beforehand.53Auction Solution Time Line:The two The two firmsfirms agree then submitto enter into sealed bids The venture isthe Auction for i11 played out.Solution ownership of The ouputat start. the venture to is available.the trust. I ITimeJV work done.The two firms The firms The bidssurrender their then play are unsealed andownership in the their venture the auction rulesventure to a strategies, are carried out totrust. C or D. find the winnerand loser and todistribute theshares and bidsaccordingly.Figure Four: The Auction Solution Time LineGiven: W = winning bid L = losing bid V total value of JV,7he winning bidder receives net payoff V - cv (W + L) - -2ihe loser bidder receives net payoff. (W + L) — -254The original game is transformed into a new game under the Auction Solution:Losing Bidder (bids L < W)C DWinning Bidder C 2c-a(W+L), a(W+L) w+s-a(W+L), a(W+L)(bids W > L) D w+s-a(W+L), a(W+L) 2d-a(W+L), a(W+L)where: payoffs expressed are not net -y/2; payoffs are read winning bidder, losing bidder;and W and L change in each cell.Assume that the auction solution uses a = 1/4; at this particular value the bids of firms matchthe total gross value of the venture19.The auction pot is made up of the value of the venturepayoff to Firm A plus that to Firm B. Consider what the bids will be in the general case whereeach firm believes the auction pot’s total value is V. If both firms bid V then a coin tossdetermines the winner, and according to the formulas, each firm receives V/2 as its “net20”payoff.Now assume it is known that one firm will bid V. If the other firm bids V+ (where ô> 0) thenit wins the auction pot and receives total net payoff of V-(2V + 3)14 < V/2 while the losing firm‘ Other values of a make firms bid differently. If a=O then each firm will bid an infinite amount as it knowsit will get its bid back and not have to transfer anything. In this case, the tie-breaking coin toss will give thefull pot to the winner and nothing to the loser. Thus, each bidder can expect a payoff of one-half the total poton average. If a= 1 then each firm will bid one quarter of what it believes will be the total value of the prize.When a= 1/4 each firm will bid what it believes will be the total value of the prize. Therefore, if there arefmancial constraints, the firms may choose to have a higher a defined in the auction rules.20 The “net” payoffs described in this section are net the costs of the auction,-y?2.55receives a net payoff of (2V+)/4 > V/2. If it bids V4 it loses the auction and receives a netpayoff of (2V4)14 < V/2 while the winning firm receives a net payoff of V-(2V-b)/4> V12.If it bids V it receives a net payoff of V/2 in either case of the coin-toss tie-breaking outcome.Now, if it is known that one firm will bid W < V then clearly the best response is a bid ofW+ ô <V in order to obtain a net payoff of V-(2W+ ô)14> V12. Similarly, if it is known thatone firm will bid W > V then the best response is a bid of W- > V in order to obtain a netpayoff of (2W-ô)/4 > V12. Thus, the equilibrium bid when a = 1/4 is V (or the true value ofthe auction pot21). It follows that in equilibrium each firm will bid the same and receive V12as its certain net payoff. When a 1/4 then bids do not mirror the total value of the auctionpot but do result in an outcome which gives a certain value of V/2 to each bidder as a netpayoff. When a = 0, an expected22 value of V12 is the net payoff to each bidder.With a set at 1/4, and knowing that the outcome of bidding will be a tie if both firms have thesame beliefs about V, the different outcomes can be analyzed. Equilibrium requires that eachfirm has true beliefs - each firm has correct conjectures over outcomes. If the outcome ismutual cooperation then each firm receives c as its certain net payoff (having both bid 2c: the21 The line of argument used to arrive at the focal bid is similar to that used to arrive at the focal price under theusual Bertrand competition case.n When c = 0 then each firm bids the same “infinite” amount and so their is a tie in the bids. When there is atie in the bids a fair coin is flipped to determine the winner. Thus, each firm will have a 50% chance of winningand receiving a net payoff of V, and a 50% chance of losing and receiving a net payoff of 0. Thus, each firmhas an expected payoff of V/2 when c = 0.Thus, if each firm knows what the outcome will be then a firm may choose to cooperate even knowing that theother will defect. If it also wins the bid, it will be able to exploit the full value of the uncompleted ventureoutput soon after t= 1, because it has no competition from the defecting firm at this time. This differs from whathappens in the original game where the defecting firm beats the cooperative firm to the market by enough ofa margin to obtain all the value of the output at t 1.56total gross value of the prize pot). Similarly, if the outcome is mutual defection then each firmreceives d as its certain net payoff. If the outcome is asymmetric (where one firm cooperatedwhile the other defected) then each firm receives (w+s)12 as its certain net payoff (see TableOne for possible outcomes).Item \ Scenario Mutual Cooperation Mutual Defection Single DefectionEquilibrium Bid 2c 2d w+sNet Payoff to each c-’y12 d--y12 (w+s-’y)12(if optimal bid wasused)riable One: Bids and Payolls for the AuctionMutual cooperation with bids at 2c is now a Nash Equilibrium24.If (w+s)/2 > d then partialcooperation is more rewarding than mutual defection. When this occurs, mutual cooperation is,in fact, the only Nash Equilibrium.24 Mutual cooperation with bids at 2c is a Nash Equilibrium because any change in either the strategy choice (i.e.,choosing to defect) or in the bidding choice (i.e., bidding above or below V) will decrease wealth. It isrelatively straightforward to show that a case of partial cooperation (where only one firm cooperates) is not aNash Equilibrium (under any potential equilibrium bid). The defecting firm can always improve its situationby cooperating instead. When payoff rankings are d > (w+ s)12 then mutual defection with a bid of 2d is alsoa Nash Equilibrium. Neither firm would change bid or strategy at this point if it had chosen to enter the auction.57The condition (w+s)/2 > d may or may not hold in general25.If (w+s)12 d then mutual cooperation and mutual defection are both Nash Equilibria.However, a forward induction argument can be used to eliminate the mutual defectionequilibrium if the ‘y-costly Auction Solution is entered into by both firms. Since the firms canassure themselves of payoff d in the original game, they would not enter the Auction Solution,play D and end up with less than d as a payof16. Thus, the only strategy a firm would playif it entered the Auction Solution is C to obtain the payoff greater than d with certainty.Therefore, the Auction Solution can be used to achieve the highest joint net payoff from theJoint Venture.In the case outlined there are no “non-transferable” costs or benefits to participating firms otherthan the investment costs which are of no strategic value. This means that there are no spilloverbenefits or opportunity costs which are individual to a firm. Thus, all payoff flows can betransferred across firms through the auction mechanism. However, in many JVPDs, non-transferable flows, such as opportunity costs that are individual to the firm, do affect the netIn a later section of this paper PD strategies are treated as investment decisions not exit decisions. Then whenthe condition (w+ s)12 > d does hold, a characteristic of the production function is revealed: The first instanceof cooperation in the venture increases the value of the total output substantially; enough to compensate for anydysfunction due to the differences in venture input choices of the two joint venture partners. For example, whenfirms have the choice of investing good versus bad engineers into the venture (as their PD strategies), thecondition translates to the good engineers being able to re-train and be quickly complementary to the bad ones.The increase in total value due to the good engineers exceeds their added opportunity cost when all costs andbenefits are summed over the two firms.26 If one firm plays D while the other plays C then they each end up with (w+s-y)12 < d by assumption. If bothplay D then they each end up with d-’y12 < d by definition. Therefore, neither firm would enter the auctionand play D.58payoffs to each individual firm and so may be of some strategic consequence. When non-transferable flows exist and have strategic implications, the auction solution may not work asjust described. This will be further detailed in Section 3.4 of this essay.The results of the analysis can now be summarized:Proposition One: If conditions exist on payoffs to firms participating in the Joint Venture:i) to make net payoffs satisfy the structure of a Prisoners’ Dilemmaand ii) to ensure that either:a) there are no non-transferable flowsor b) the non-transferable flows are of no strategic consequencethen implementing an ex-ante auction (of the type described here) over thetotal gross payoff of the venture will result in the highest joint net payoffoutcome of that venture.Proof: As shown above, when an efficient auction is implemented the only NashEquilibrium that survives forward induction is one of mutual cooperation in thePD itself. When this occurs, the optimal outcome of the Joint Venture is realised.059To this point the scenario has been one where there are two firms, the firms are symmetric, andthe non-transferable costs are eliminated or are of no strategic value. There are some veryinteresting results that occur when any of these restrictions is relaxed.3.2 An Application with n >2 FirmsThe extension to n >2 firms is straightforward. Consider the basic case with symmetric firms.The highest bidder obtains full ownership of the venture and its own bid back less the sum ofthe cr-shares of the total bids collected. The losing bidders get their bids back plus the cr-shareof the total bids collected transferred from the winner. If cx is set to equal 1/n2 then the bids willagain reflect the gross total value of the JV.Under the Auction Solution each firm plays the cooperating strategy and bids nc for the n sharesof the venture. The tie is broken by an appropriate randomizing devise. Each firm ends up withc as its net payoff. This is the dominant outcome as long as the PD structure remains as definedwith w > c > d > s and nc > w+ (n-1)s. A forward induction argument is used when thecondition w+ (n-1)s > nd is violated based on a ‘y-costly auction. The mathematics for thebidding and the Nash Equilibria are derived in a straightforward manner from the base case ofthe Auction Solution described in the previous section.603.3 An Application with Asymmetric FirmsWith symmetric firms the basic auction described in Section 3.1 has much in common with usinga simple coin-flip for ownership of the JV. The similarity is much reduced when firms are notsymmetric.A different (but consistent) approach for fmding Nash equilibria is required when the firms arenot symmetric as the strategy space is more complicated.But first it is necessary to define what is meant by asymmetric firms. Assume that Firm A isbetter able to capitalize on the JV’s outputs so that it receives a higher gross payoff than FirmB in any of the outcome scenarios, (i.e., CA> CB, WA> wB, dA> dB and SA> s with the conditionsthat WA> CA> dA>sA, WB > CB> dB> SB, and CA+CB+CB+CA> WA+SB+WB+SA holding tomaintain the PD form27). Assume this is common knowledge. Assume also that the investmentcosts stay the same (and so are the same for each firm). Imagine that the added benefit to FirmA in each case comes from some added spillover from the venture output that can be onlyexploited by Firm A due to its control of certain complementary resources or such. Non-transferable payoffs are involved here (i.e., the difference in value between CA and CB, betweendA and dB, etc..) that are not simply contingent on which PD strategy each firm played (i.e.,Firm A will always obtain some non-transferable payoff if it wins the auction regardless of the27 Thus, if both firms cooperate, Firm A gets CA and B gets cB; while if both defect A gets dA while B gets d,and so on for the other outcomes. It is assumed that the increased reward that A receives over B for any of thepayoffs w,c,d or s carries over to when A obtains ownership of the venture. For example, if A won ownershipwhen both firms cooperated in the venture then A would receive 2CA (and not CA+CB)as its gross payoff fromthe venture.61strategy played by Firm B).If A wins ownership of the whole JV, its gross payoff would be 2CA if A and B cooperated,WA+SA if only A or B cooperated, and 2dA if both A and B defected. Similarly, if B winsownership of the whole JV, its gross payoff would be 2CB if A and B cooperated, WB+SB if onlyA or B cooperated, and 2dfi if both A and B defected.Under the auction, if A wins ownership by bidding appropriately28,the venture is transformedinto the following:Firm BC DFirm C 2CA-CB, CB wA+SA-(wB+SB)/2,(wB+sB)12A D WA+SA-(WB+SB)/2,(wB+sB)/2 2dA-dB, dBThe socially efficient outcome would entail Firm A obtaining full ownership rights to a venturethat produces the highest joint outcome (that which arises from mutual cooperation). Specifically,the socially efficient outcome would entail Firm A winning the auction when the AuctionSolution is implemented.28 In the table shown, the appropriate bids are composed of the total venture value to B in each case, plus aninfinitely small increment.62Consider now what does occur when the firms are asymmetric and there is full information. Theauction works efficiently when 0 < a 1 (if a = 0, then the solution is simply a coin-flip todetermine ownership of the venture). Assume a = 1/4 as before. Now check for the optimal bidfor the lower value firm, Firm B. In the general case in which it believes the auction pot’s valueis V, if Firm B bids V+ ô (where ô >0) while Firm A bids only V then B wins the auction potand receives total net29 payoff of V-(2V+ )/4 < V/2. If B bids V-ô while Firm A bids V thenB loses the auction and receives a net payoff of (2V4)14 < V12. If B bids V while Firm A bidsV also then B receives a net payoff of V12 in either case of the coin-toss tie-breaking outcome.Therefore, the optimal bid for Firm B when Firm A bids V and a = 1/4 is V (or the true valueof the auction pot for Firm B). Now check what the optimal bid is for Firm A.It has been shown that if Firm A also believes the auction pot’s value is V then its best bid isV. In fact, even if Firm A believes the auction pot has a value above V to itself, then a bid ofV will ensure it an expected reward of the average of V and that higher value. If it bids V+while Firm B bids only V then it wins the auction and realizes a total net payoff of V+fl(2V+ c)14 (where j3> € is the added value Firm A has of the venture outcome over that of FirmB). This is larger than the payoff from simply tying the auction, (V+fl)/2. Now recheck theoptimality of Firm B’s bid in light of Firm A’s optimal bid. If Firm B bids above A’s bid thenB receives a total net payoff below V12. If B bids below A’s bid, so that B bids V, then Breceives a total net payoff of V12 + d4. If B bids the same as Firm A at V+ E then B ties andAgain, the “net” payoffs described in this section are net of the auction costs, 7/2.63receives a total net payoff if it wins the coin-flip of value (V-e)12 and if it loses of value(V+ c)12 for an expected value of just V/2. Therefore, the optimal bid for Firm B remains at V.It is conjectured that the Auction Solution will achieve the efficient outcome in which A winsthe auction (with a bid of V+ , where V equals2cB), and mutual cooperation occurs. It has beenshown that this outcome is possible in bidding, now it must be shown that it can occur in gamestrategies played.When both firms cooperate and bid as predicted above then Firm A gets2CA-CB-C/4and FirmB gets CB+ d4. If B could obtain a higher payoff by defecting, it would. To rule this outrequires, first, that WE+SB-CB-3/4< C+c/4 so that defecting and bidding to win the auction(when A is expecting cooperation) is less attractive than cooperating and losing the auction30.When this restriction is met, B is indifferent between cooperating and defecting but will bid asrequired31.° Of course, if B defected and bid to lose the auction (when A was expecting cooperation) B would still obtainCB+ €14.31 Although B may be indifferent in the case described so far, some considerations that are outside the basicmodel, like reputation effects, may make B partial to cooperation. Even if B could not obtain a higher payoffby defecting and winning the bid, B may need to have a disincentive to defecting and losing the auction. B couldobtain the same payoff if it bid the same and defected while A cooperated and bid to win. Therefore, a wayto move B off the indifference point (of cooperating or defecting) may be required here. It should be noted thatthe indifference is generated by the fact that B knows it will lose the auction and so does not care about itsstrategy, as its reward is not affected by that choice directly. Therefore, what is needed is to create someuncertainty for B over its losing the auction in order to give it the incentive to cooperate. This is done simplyby having A randomize its bid between 2c5 and 2CB+ e, with a very small but positive probability on biddingjust 2c5. When this occurs, B wants to bid 2c5 and play C for a minimum payoff of c5. If it choose either tobid less, or to play D, it would ensure itself that it would obtain a smaller payoff with positive probability. Thisis, in effect, a way to show that B will cooperate under the Trembling Hand equilibrium concept.64Given B’s strategy, A may have an incentive to change strategies. If A defected and bid thesame as before, A would obtain wA+sA-(4cB E)/.Thus, if WA+SA < 2C, then A would notwant to change strategies.Therefore, to obtain the outcome where the firm with the highest valuation of the venture winsownership and mutual cooperation is a Nash Equilibrium of this PD under the Auction Solutionrequires that two payoff rankings hold that are not guaranteed by definition of a PD:Require both 2 CB > WB ÷ SB as well as 2 CA > WA +for obtaining the desired outcome, althoughknow that 2 CA + 2 CB > WA + 3A + wE + SB giventhe definition of the PD.The required payoff rankings would be met when like firms are involved. The first payoffranking (i.e., inequality) is met when two “B” firms are involved in the JVPD. The secondpayoff ranking is met when two “A” firms are involved in the JVPD. The third line shows thepayoff ranking that is met when two unlike firms are involved in a Prisoners’ Dilemma (as isthe case here - both an “A” firm and a “B” firm are involved). It should be noted that the PDcharacteristics do not arise due to the presence of unlike firms, but occur due to the nature ofthis JV.Now, assume the required payoff rankings hold and analyze this game for equilibria. Partialcooperation (where only one firm cooperates) is never a Nash Equilibrium. At this point, thedefecting firm will always want to change its strategy (and its bid) to obtain a higher payoff,65given both the payoff ordering requirements above hold.The only other candidate Nash Equilibrium to check is that of mutual defection. If both defectand A bids 2d+ c while B bids 2dB then Firm A gets 2dA-dB-c14 and Firm B gets dB+ €14.Firms may want to change their choices at this point depending on the payoff ranlcings. If 2db,> ‘.v+S then A has no incentive to change its choices. If 2dB > WB+SB then B has noincentive to change its choices. When these two conditions hold, if the firms are in the auction,then this mutual defection point is another Nash Equilibrium. However, as long as €14 < ‘y12,which will hold true for very small€, B has no incentive to enter the Auction Solution anddefect. As Firm A knows this, and does better by cooperating than defecting (under the payoffrankings assumed above), then A will also cooperate. As B knows this, and obtains a payoff ofCB by entering and cooperating instead of d by not entering, B will enter, and then so will A.Thus, the mutual defection Nash Equilibrium is ruled out by forward induction.The only Nash Equilibrium that survives forward induction is that of mutual cooperation (underthe payoff ranking assumptions above). Therefore, the Auction Solution is an efficient methodof solving the PD under certain restrictions in the case of asymmetric firms. It should be noted,however, that these restrictions are not automatically satisfied by the definition of the game asit has been presented32.When these restrictions are not satisfied then mutual cooperation is no32 The conditions would be met if it is assumed that the added spillover to Firm A is non-decreasing as the ventureoutcome becomes more valuable (becomes more complete). Thus, the added spillover when there is at least onedefection is the same as when there are two as the venture is terminated at the end of period one in either case.It is assumed that that added spillover is not greater when there are no defections. Therefore, this gives thefollowing conditions: WA+SAWBSB = dA+ dA-da-da and CA+CACBCB dA+ dA-da-da. Thus, a monopoly ona more valuable product is worth more than one on a less valuable (incomplete) product. From the two66longer a Nash Equilibrium because at least one firm has an incentive to defect on the other’scooperation.A case in which these conditions are satisfied arises when the difference between Firm A’spayoff and Firm B’s is constant over the outcome space: CA=CB+A, WA=WB+A, dj=dB+A andSA=SB+A. The original PD game definition now becomes4cB>2(w +sB). Now the AuctionSolution for these asymmetric firms is applicable as the two restrictions are met. The solutionwill also work under similar restrictions when it is not known by how much the higher-valuingfirm values the venture (as such information has not been required by the solution presented herethus far)33.Now consider the effects of a less complicated solution technique on the case of asymmetricfirms. This solution is an extreme member of the Auction Solution set; it occurs when X is setto zero. This amounts to a coin-flip to determine ownership of the venture. It shall be termedthe Coin Flip Solution. The firms again place their shares of ownership in trust to be held untilequations above it follows that: 2CA> WA+SA and 2c8> w5+s. When these conditions are met, the Payofforderings required under the general case of asymmetric firms are met, and the Auction Solution achievesefficiency.Consider another case of asymmetric firms, where each firm draws from a distribution (i.e., the samedistribution for both players) of spillover capabilities which affect payoff evaluation. Each firm knows thedistribution but not the other firm’s draw. Although this is a case for future work, one possible outcome toconsider would be that each firm would (usually) bid its true valuation of the JV output and receive a certainpayoff of one-half its own true valuation of the total auction pot plus one-quarter of the absolute value ofdifferential in the values (when a= 1/4). This outcome is more efficient than using a simple coin flip todetermine ownership of the venture (or when a=O). The coin flip results in less expected total payoff, as it isnot the best firm which wins the auction each time:3V+V V+VAuction joint payoff is A B versus A28 for the coin flip.67the venture ends. Before the venture begins (or upon completion) a fair coin is flipped and itsresult left secret until the venture ends whereupon the outcome of the flip is revealed. At thattime the full ownership is handed over to the winning firm. Under this solution each firm, ineffect, will have an expected payoff of half its valuation of total ownership of the venture.As in the original Auction Solution when firms are symmetric and risk neutral they willparticipate in the Coin Flip Solution and cooperate when both c> (w + s)12 (which is met bydefinition) and (w+s)12 > d. When (w+s)12 < d, then a forward induction argument can beused to eliminate the Nash Equilibrium of mutual defection.This solution appears to be less complicated than the Auction Solution (as bids are not submittedor counted). However, it does not come up with the optimal solution in the case of asymmetricfirms. Only with expected probability of 1/2 will the ownership of the venture go to the firm thatvalues it more highly.The Coin Flip Solution is more formally presented in Appendix Eight and shown to bedominated by the Auction Solution when firms are risk averse.‘ If a firm enters and defects then it either gets (in expected value terms) d-I2 or (w+ s.-y)/2 which are both lessthan d which it can obtain with certainty by not entering the Coin Flip Solution. Again, it has been assumedthat this technique for resolving the PD entails a cost. The cost, y, can be assumed to be zero in which caseforward induction does not eliminate the partake-in-the-solution-and-defect equilibrium.683.4 An Application with Strategic Non-Transferable Costs (SNTC)35The case of asymmetric firms provides a natural introduction to SNTC as these are in essenceasymmetries of a different type. While the asymmetry just described arises due to differencesin payoffs to each firm independent of who played which PD strategy, the asymmetry that arisesdue to SNTC is due only to who played which PD strategy (i.e., who defected and whocooperated). In the SNTC case, the asymmetry arises due to cost differences between playingD versus playing C and so it does matter to each firm who defected and who cooperated.The non-transferable amounts are extra costs and benefits that cannot be transferred betweenparticipating firms by redistributing the shares of the venture. For example, the opportunity costthat a participating firm incurs by sending an engineer to work in the venture is an investmentcost that it will not be directly reimbursed no matter who ends up with ownership of theventure36.While the valuable new product or process can be transferred in ownership rightsto the final owner of the shares, the non-transferable extras (like the opportunity costs or addedspillover benefits) cannot. These extras are in essence outside the venture itself but do affect thenet payoffs of the participating firms.Now, consider the case of two symmetric firms participating as equal partners in an R&D JointVenture. They can cooperate by investing in the IV the time of good engineers and machinesAnalysis of any and all combinations of the three derivations from the basic application (as described in Sections3.2, 3.3 and 3.4) is not done in this essay but left for future work.36 The firm will be indirectly compensated for its investment when it receives whatever reward it gets from theventure. However, the choice of the quality of the investment alone does not guarantee proportionally greaterrewards for a firm. This is partly because a firm’s marginal cost and benefit are not those of the venture.69at cost i or they can defect by investing in the JV the time of bad engineers and machines atcost‘D where ‘ > 1D (assume i= 0 without loss of generality). It is implied by this inequality thatthe opportunity cost of sending bad resources is less than that of sending good ones. Further,assume that verification by a third party of which investments were made is too costly37. Thedifference in opportunity cost between investing the time of good engineers and bad onesrepresents a private saving to the defecting firm while the cost of the defection is not totallyprivate.Assume that the output of the JV increases with better investment so that mutual cooperationresults in the highest gross JV output (P), single defection the second highest (PcD), andmutual defection the lowest (PDD; such that CC >P> DD >0). Now define the net payoff perfirm from mutual cooperation as c = (PI2)-i; the net payoff from mutual defection as d =1DIi2; the net payoff from defecting while the other cooperates as w = CD’2; and the netpayoff from cooperating while the other defects as s = (PcD/2)-L, . In order to be a PD, payoffsneed to be structured: w > c > d > s and 2c > w+s. (In this case, d > 0 is assumed so thatfirms willingly participate in the JV.)Verification by a third party of whether a firm cooperated or defected may be too costly or unattractive toparticipating firms for a number of reasons. For example, it may be very difficult without intimate knowledgeof a firm which of its engineers are relatively good versus bad. Also, it may be unattractive to firms who areengaged in a venture that produces valuable patents or trade secrets to have outside parties become aware ofthem before they can be fully protected.70The original game:Firm BC DFirm C (PI2)-i, (P/2)-i (PcD/2)-ic, CD’2A D CD’2, (PcD’2)-1c ‘DD”2,DD’2is transformed into the a new game under the Auction Solution:Losing BidderC DWinning C Pcc-ic-(bA+ bB) /4, (bA-I- b)/4-i PcD-ic-(bA+bB)/4,(bA+ b)!4Bidder D PCD-(bA+bB)/4,(bA+bB)/4-1C PDD-(bA+bB)/4,(bA+bI)/4where: bA is Firm A’s bid and bB is Firm B’s bid and these bids can differ in each cell;a= 1/4; these payoffs are not net the auction set-up cost -y/2; and the winningbidder has a higher bid than the losing bidder.The game is a Prisoners’ Dilemma with only one Nash equilibrium: mutual defection. Thisgeneral description of a JVPD involving SNTCs differs from the first JVPD description that didnot involve SNTCs. Under this new JVPD the simple Auction Solution alone will not work. Thisis proven by example.71Consider the venture as defined above with the auction technique as defined previously and withset to 1/4. To prove that the Auction Solution fails to elicit mutual cooperation it is sufficientto show that mutual cooperation is never a Nash Equilibrium. Assume that mutual cooperationis the conjectured outcome at bids bA and bB (for Firm A and B respectively). If a firm defectedinstead and bid CD, it would ensure itself a payoff of at least w > c when the other firmremained cooperative, regardless of that other firm’s bid. If the cooperative firm bid ormore and won, the defecting firm would receive at least (Pcc+PCD)/4,or (w+c+i)/2, whichis larger than c. If the cooperating firm bid less than P but greater than or equal to CD andwon, the defecting firm would receive at least CD’2 = w > c. If the cooperating firm bid anyless it would lose the auction, and the defecting firm would receive at least CD’2 = W > C.Therefore, each firm has an incentive to at least change strategy at this point of mutualcooperation. Mutual cooperation cannot be a Nash Equilibrium under the Auction Solution asdefined.Mutual cooperation is no longer a Nash Equilibrium as it was when there were no SNTCs. Thisis because with SNTCs, a firm that defects can receive a higher payoff than the firm thatcooperates in equilibrium. When there are no SNTCs, both firms receive the same amount inequilibrium so it does not pay to defect. When SNTCs exist, the defecting firm can assure itselfof payoff w (with an appropriate bid) and therefore obtain more than the cooperating firmregardless of which firm wins the auction. The difference in payoffs between the cooperatingand defecting firm is composed of the SNTCs.72It has already been shown that the mutual cooperation point (under any bids) is not a NashEquilibrium. It is also straightforward to show that mutual defection under the Auction Solutionis not an equilibrium. If bids are above or below DD then one firm can do better by changingits bid to approach PDD. For example, if bids were tied at PDD+ E then Firm A could reduce itsbid to P and increase its payoff by c14. Thus, it is sufficient to prove that mutual defectionwith bids of DD is not an equilibrium. If w+s >2d then each firm has an incentive to changeits choices to cooperate and bid DD + € in order to obtain a payoff of w + s-d-€14-y/2 which isassumed to be larger than d-yI2. If w+s 2d then mutual defection with bids of DD is a Nashequilibrium f the firms entered the Auction Solution. However, they would not enter theAuction Solution as they could increase wealth by y/2 through not entering.All that is left is to show that partial cooperation is not a Nash equilibrium under the AuctionSolution. If bids are above or below CD then one firm can do better by changing its bid toapproach For example, if bids were tied at PCD-€ then either firm could change its bid toCD, win the auction, and increase its payoff by El4. Therefore, assume that the bidding isconsistent with the strategic outcome; bids focus on CD At this point, the cooperating firmreceives s only and, therefore, has an incentive to defect instead and bid a minimum of pD toensure itself a minimum payoff of d > s. Thus, partial cooperation is not a Nash Equilibrium.Therefore, there are no pure strategy Nash equilibria under the Auction Solution (regardless ofthe bids played). The only possible Nash equilibria to the JVPD left to explore then are: notentering the auction; or, entering the auction and playing mixed strategies and bids. When73payoffs are ordered such that no mixed strategy Nash equilibria entail both firms receiving morethan d as a net payoff, then not entering the auction is the only Nash Equilibrium38.If payoffsare ordered otherwise, then firms are better off entering the Auction Solution and playing mixedstrategies and bids. As playing mixed strategies implies cooperating with positive probability,the Auction Solution does involve a Pareto improvement (under the payoff ordering assumed)39.However, the (near) jointly efficient outcome can be restored under the existence of SNTCs witha redefined auction. If the choice of bids is restricted to the three focal amounts then the AuctionSolution can be efficient. The focal amounts are those defmed by the three possible ventureoutcome payoffs: 2c, w+s, 2d (these particular amounts are based on symmetric firms and a= 1/4). Now assume that firms can only choose (and be held to only) one of these three bidswhen participating in the auction.Analyzing this new case involves checking all possible strategy combinations. The firms can playpure or mixed bids (the bids that are mixed correspond to the two possible outcomes that couldFor example, consider mixed Nash equilibria when one firm always defects while the other randomizescooperating and defecting, and assume that the firms bid efficiently based on their beliefs and knowledge. Thiscase defines the worst possible situation where the firms have a chance to be more profitable by entering theauction (i.e., if both firms randomized cooperating and defecting then the chances of higher payoffs wouldgenerally be greater). Straightforward calculations reveal that for both firms to receive at least d as their netpayoff requires the necessary condition that w+s > 2d. This may be considered as one of the payoff orderingsthat must be met in order to have a Nash equilibrium that involves partaking in the Auction Solution.As the choice of bids is infinite, it is generally complicated to find the Nash Equilibrium in mixed strategiesand bids when the firms do enter the solution. However, the following example shows that a Paretoimprovement can occur at such equilibria: Assume that the payoffs are ordered such that: w+s-y > 2d andalso c+ d = w+ s. There is a Nash Equilibrium in mixed strategies when firms cooperate half the time andalways bid w+s. Each firm’s expected payoff is then (w+s--y)/2 which is a Pareto improvement under theassumptions.74occur when one firm assumes the other will play a certain strategy). The analysis (seeAppendix Eleven for details) gives a condition on IC to ensure mutual cooperation and mutualbids of 2c:2 c — (w + s) > ‘When this condition holds (which is not guaranteed by PD definition or by the assumptions) thenmutual cooperation is a Nash Equilibrium. It is the only Nash Equilibrium if a further conditionon IC holds which ensures that a firm will cooperate on another firm’s defection:(w ÷ s) - 2 d> 4 iHowever, when 2d- (w+s) + IC13 > 0 then mutual defection (with consistent bids of 2d) isanother Nash Equilibrium jf the firms enter the solution. This equilibrium can be ruled out byforward induction. No firm will partake in the solution4’and defect as it would ensure itselfa dominated payoff’2.° For example, if Firm A is trying to find its best responses assuming that Firm B will cooperate, then it has toassume that B will randomize bids between 2c and w+s.41 Firms will only enter the auction under forward induction if there is some possibility that they could improvetheir payoffs, i.e., if C > d+’y12.42 For example, by not entering the solution, it would get a payoff of d instead of a possible payoff of d-I2 byentering the solution.75Thus, under a slightly redefined Auction Solution43 the Pareto-optimal outcome of the JVPDcan be obtained (given the parameters meet certain restrictions) when SNTCs are present.3.5 Solutions Under General SNTCsThere is still a way to solve the PD even when SNTCs are present but the parameters do notmeet the required restrictions. It requires certain further assumptions and an additionalmechanism. The mechanism required is a special futures-like contract on each participant’sstock. The assumptions required are for near perfect stock markets and near perfect accountingdisclosures.The special futures-like contract would be created and given only to participants in the jointventure. It is issued at the time the joint venture is announced and only if the JVPD Solution isin effect. Its maturity date is the completion of the joint venture. The contract entitles the holderto the difference in the worth of the stock between the maturity date and the issue date45.Note that the simple coin toss for ownership version of this solution will not work. The bidding space cannotbe restricted when a is set to zero, so that a value is ruled out in the redefined Auction Solution.The mechanism and assumptions are possible in this case where Prisoners’ Dilemma participants are companiesand not people (as there is a working market for stocks of companies but not for people as of yet).‘ When the difference in worth is positive, the holder has the incentive to redeem the contracts. When thedifference is negative, the issuer can force the holder to redeem the contracts and pay the difference. Theconditions on the contracts are not as restrictive as they may first appear. The issuer can offer to buy these fromthe holder at any time. This means that after the strategy has been chosen and before the maturity date, a firmmay be able to get back its contracts if it had other (privately known) projects to do and did not want the otherfirm profiting from the increase in contract value.76The assumption that the market is near perfect means that the only difference in the stock pricethat the special futures contract will capture is that company’s net payoff from the joint venture(which would include the opportunity cost of the company and any spillovers that the companywould capture). Therefore, all information regarding the initiation of the venture would beprivate to the firms involved and no inside information or other leaks are assumed to occur thatwould allow the market to anticipate the transaction and alter the value of the special futurescontracts. The assumption that the accounting disclosure is near perfect ensures that the stockprices will accurately reflect the net value of participation in the joint venture; therefore, noinformation is hidden from stockholders.The mechanism and assumptions described allow previously non-transferred payoffs of theventure to now be transferred. For example, now opportunity costs of participants can betransferred by having the other participants hold each others’ special futures contracts.When the payoffs of participation in the joint venture are transferrable or are made transferrablethen the game is transformed into one where maximizing joint payoffs is in line with maximizingindividual payoffs.There are many different types of solutions (see Appendix Nine for details) to the SNTC JVPDwhen the special futures contracts are available, including a new Auction Solution.773.5.1 The New Auction SolutionIt is assumed that participating in the joint venture generates some costs or benefits which accrueto the individual firms above those accounted for in the net value of the joint venture itself.Those costs and benefits must be addressed as they are non-transferable (as is assumed in thecases of SNTCs). It is assumed that these individual costs and benefits can be made transferableby issuing special futures contracts as described above. The special futures contracts cover thefull stock of each participant and are put into the auction pot (along with the shares of theownership of the Joint Venture) to make up the total prize.Thus, at the same time that the rights to ownership of the Joint Venture are placed in trust soare the special futures contracts. The rules (and costs) of the original Auction Solution apply.When c set at 1/4, each firm will bid the true value, V, of the new total auction pot.With ce set at 1/4 in a -y-costly auction, and knowing that the outcome of bidding will be a tieif both firms have the same beliefs about V, the different outcomes can be analyzed. Assumethat in equilibrium each firm has true beliefs. Therefore, in equilibrium, each firm holds thesame beliefs and those beliefs correspond to the outcome being analyzed. If the outcome ismutual cooperation then each firm receives c as its certain net payoff (having both bid 2c -the total value of the prize pot). If the outcome is mutual defection then each firm receives d asits certain net payoff. If the outcome is asymmetric (where one firm cooperated while the otherdefected) then each firm receives (w+s)12 as its certain net payoff.46 These payoffs are not net the cost of setting up the auction, ‘y/2.78As with the solutions presented before mutual cooperation is a Nash Equilibrium47,and the onlyone that survives forward induction, given certain conditions. If (w+s)/2 > d then mutualcooperation is the only Nash Equilibrium. If (w+s)12 d then mutual cooperation and mutualdefection are both Nash equilibria. A forward induction argument based on the added cost ofthe Auction Solution can be used to eliminate the mutual defection equilibrium if the AuctionSolution is entered into by both firms48. Therefore, the New Auction Solution can be used toachieve the highest joint net payoff from the Joint Venture if SNTCs and special futurescontracts are present.The solution considered thus far in this section (and in Appendix Nine) entails -y-transactionscosts. It is a Pareto-improving application if the reward to each firm net the solution set-up cost,‘y, is greater than d (which has been assumed to hold).In Appendix Ten, this solution is compared to the others (from Appendix Nine) and someextensions to these solutions are analyzed to show their robustness.As in section 3.1 it is the case that neither finn would have any incentive to change either its bid or its PDstrategy at this point.If a = 0 then a forward induction argument, based on expected values versus certain values if players are riskaverse, can be used to eliminate the mutual defection equilibrium instead.794. The Contracting AlternativesHowever, it is difficult to judge the merits of these solutions without an analysis of the usualpossible alternatives used to solve such dilemmas. This analysis is carried out in the followingsection.There are other alternatives to the solutions presented above that may also provide an efficiencyimprovement to the JVPD but have different implementation requirements and costs. The Coin-Flip Solution (see Appendix Eight) may be unattractive due to the risks (of losing) involved. TheAuction and Transference Solutions (see Appendix Nine) may appear to be quite complicated.However, all of these solutions hold one main advantage over the contracting alternatives in thatthey do not require any third-party verification of the PD strategies (i.e., C or D) played. Eachfirm enters into these solutions knowing that the JVPD outcome is what is at stake and that howit resulted is irrelevant to their payoffs. The payoffs to the firms are not changed by knowingwho cooperated and who defected ex-post. Therefore, no third-party verification is desired norrequired in any of these three solutions.The contracting alternatives, however, do require the third-party verification. The payoffs to thefirms are based on proving who cooperated and who defected ex-post. The alternatives thatrequire this third-party verification are full contracting, and side payment arrangements. Thesealternative solutions can provide a Pareto-improvement to the JVPD if their implementation costsare low enough. Once again, if the payoff to each firm net implementation costs is greater than80d, then these solutions are attractive. However, the costs of these contracting solutions maydiffer from the costs of the Coin-Flip, Transference and Auction Solutions as third-partyverification is now required.Under contracting where, for example, the two firms would define ex-ante the arrangement togenerate the mutual cooperation outcome, ex-post third-party verification of the strategies playedis essential. This is because each party has an incentive to maximize its own payoff and noincentive to maximize the joint payoff. The contract may specify some penalty for breaking theterms sufficient to ensure, under perfect ex-post strategy verification by a third party, anadequate incentive to play the cooperative strategy. Thus, two third-party verifications arerequired (at a minimum) as each firm must be able to verify that its own play was correct to theother so as not to have a penalty imposed upon it. The efficient penalties involved would besimple transfers between affected parties. If both defected, there would be no transfer. If onlyone firm defected then there would be a transfer to the cooperating firm of a sum sufficient tooffset its loss from a normal PD outcome (i.e., transfer = d - s) plus perhaps some additionalamount to make it better off than the defecting party. In any case the contracting alternativerequires some additional third-party verification and possible material expense and complicationcompared to the Coin Flip, Transference and Auction Solutions.There are also two main side-payment arrangements to consider as alternative solutions. The firstarrangement is one in which one firm offers some side payment, , to the other firm if it playsC in the game. Consider that the offer is only “enforceable” on the offering firm: if the offeree81can verify that it played C, then the offerer’s payment is enforced. The offeree is not forced toplay C; and if it does not play C, the offeree is not forced to pay & Compared to the contract,there are no penalty terms and only one third-party verification is required.Now consider the side-payment requirements, the firms’ actions and the outcome. When theofferer initiates the scheme, it does so to maximize its payoff. Inspection of the normal form ofthe original game shows that this occurs when it plays D and the other plays C. Thus, theofferer pays the offeree ô to play C so it can play D and receive w-3 as its net reward. In orderfor it to be better off than in the original game this reward must be greater than d. For theofferee, it will only accept the offer if it does better than in the original game and if it doesbetter than breaking the offer contract; in either case it plays C if its net payoff of +s is greaterthan d. These conditions require: w-d> ô> d-s or that w + s > 2d. This requirement is not arestriction of the PD game. Therefore, this solution may not be available in all JVPD situations.If it were, however, who would want to be the offerer and who the offeree would depend on thevalues of the two net rewards, w-3 and +s. The payoff to each would be equal when the offeris equal to 5= (w-s)12 (which would be the case under a Nash Bargaining Solution between theparties). The payoff to each firm is then (w+s)/2 (and this is better than d when the restrictionof w+s > 2d is met). Under this side-payment arrangement the total gross welfare generated isw+s which is less than the gross under the other Solution alternatives (e.g., the AuctionSolution). However, this alternative is relatively simple but does require one third-partyverification. If the restriction on payoffs is met and the costs to implement this solution lead tonet payoffs to each firm greater than d then implementing this side-payment arrangement is82beneficial.The second side-payment arrangement occurs with similar one-sided “enforceability” but alsorequires that payoffs be transferable. A further Pareto-improvement is possible in this scenario.Consider one firm offering to the other firm if that other firm plays C and transfers to theofferer its share of the JVPD net payoff. In order for the offerer to maximize its payoff in thisscenario, it will also play C to maximize the total payoff to be split in this scenario. The resultis that the offerer receives a payoff of 2c-ô while the offeree receives 5. The offerer will havean incentive to offer only if its payoff is better than in the original game - if 2c-ô> d. Theofferee will only have an incentive to accept if its payoff is better than in the original game - if3> d - and if it cannot do better by breaking the offer contract by defecting knowing the offererwill cooperate - if ô> w. These requirements place a restriction on the offer that 2c-d> b> w.This places a restriction on the PD payoffs that 2c> w+ d which is not a requirement of the PDgame definition. If, however, this restriction is met then by inspection it can be seen that theofferee does better than the offerer. Therefore, no one may want to offer and another Prisoners’Dilemma results. If, however, this arrangement is implemented then the total gross welfaregenerated is 2c which is as high as that generated by either the Transference, Coin-Flip or theAuction Solution. However, this arrangement, while somewhat straightforward, does requirecosts in the form of one third-party verification and some contracting. If it can be implementedand the net payoff to each firm is greater than d then this arrangement is an attractive alternativeto consider.83While these alternative solutions require some third-party verification, the Auction Solution doesnot49. The reason that this is important is that the cost of third-party verification may berestrictive. The cost of the act of third-party verification may also be a substantial as thejudgement on whether a firm cooperated or defected may be based on a relative measure,relative to the firm itself. In these cases, then, where third-party verification is very costlyrelative to the other set-up costs (and relative to the payoffs generated from the joint venture),these contracting alternatives are inferior to the Auction Solution (original or New).5. ConclusionsThis essay has presented implementable solutions to a JVPD - the Auction Solution (original andNew and those other Solutions found in Appendix Nine). Even when the payoffs are notcompletely transferable, an appropriately constructed Auction Solution may result in a Pareto-improvement to the Joint Venture under certain parameter ordering restrictions and theavailability of certain futures markets. Therefore, policies that enable the Solutions to beimplemented are encouraged when any of these Solutions is the best way to solve the dilemma.The Auction Solution has many advantages. The auction mechanism is simple to understand,legal, and requires few resources (just a machine to hold bids, compare them, and then distributethe shares and bids). It allows the optimal bidder (the one with the highest valuation of theNeither does the Transference or the Coin Flip Solution; see Appendix Nine for details.84venture) to obtain ownership. The Auction Solution may also be more acceptable undercompetition law. It does not have the same “overly-cooperative” appearance as contract-basedscenarios (like the side-payment solution); after all, the auction is a competitive one. TheAuction Solution also appears to have the most flexibility for obtaining the cooperative outcometo any joint venture that can be represented by a Prisoners’ Dilemma, especially if it entailsstrategic non-transferable costs.The solutions presented in this essay entail some strong assumptions to be workable. It isassumed that in most cases that there are no non-transferable costs involved in the JV, and thatmany JVs exhibit the form of a Prisoners’ Dilemma. As well, it is assumed that it is relativelyeasy and inexpensive to implement the solutions.Future work may include relaxing some of the assumptions. It may also include experimentaltrials of the solutions on subjects. Perhaps even empirical work may be attempted where buyoutsof the JV by one of the partners will signal that an auction took place.Even taking the assumptions as given, the model has presented some worthwhile solutions topossible dilemmas. The Auction Solution has even been shown to be a viable method that wouldenable some JVs to be undertaken that would not be otherwise (i.e., when c > 0 but d < 0).There may be even further benefits to consider when the cooperative outcome to the JVPD isattained. Kamien et al (1992) study welfare effects of some Research Joint Venture types. They85find that cooperative Research Joint Ventures (cooperating in the R&D decisions) result in thehighest social welfare compared to independent or competitive R&D ventures. The findingreveals that there may be additional economic benefits to obtaining the cooperative (C,C)outcome in R&D Joint Ventures above that specified by the participants in the venture itself.There may also be positive consumer surplus effects and other spillovers to consider as, forexample, occurs under the original venture scenario when the venture output is allowed to becompleted and offered to the market.86CHAPTER THREE: ESSAY THREEFISH AND SHIPS: TRADE WITH IMPERFECT COMPETITION ANDAN INTERNATIONAL OPEN ACCESS RESOURCE1. IntroductionThis essay considers the problem of trade between two countries that share a very valuable openaccess resource. Addressing this problem may be increasingly important as the world approachesthe limits of some key shared resource stocks. The problem is modelled both as a one-stagetrading game where nations are price-takers and a two-stage trading game where nations assumemarket power. The one-stage game yields the familiar Ricardian outcome when the amount ofthe open access resource is not too low. The two-stage game yields a different outcome (i.e.,division of welfare) and so can be considered as an alternative way of modeffing the problem1.It also offers a potentially attractive method of structuring trade for the nation who stands tobenefit from the new division of welfare.In the absence of market failures, such as those caused by externalities and imperfectcompetition, trade is welfare-improving for the participants. However, the gains in welfaregenerated by trade may be substantially decreased by losses resulting from the negativeThe two-stage game may explain to some extent why nations who trade do not obtain the Ricardian result. Itmay also explain why trade may be unattractive when there are non-zero costs which are not compensated forby expected Ricardian gains from trade. The typical Ricardian result would entail at least one of the two nationsspecializing fully in the production of one of the intermediate goods. This type of trading is outlined in mosteconomics texts such as Krugman and Obstfeld (1991).87externality arising from an open access resource. When the open access resource is scarceenough that its successful exploitation is measured by a nation’s relative fitness to harvest thatresource, trade may alter such relative fitness to the detriment of at least one nation. Similarly,when nations exercise market power in imperfectly competitive markets, the welfare gains fromtrade may be substantially decreased. When both sources of market failure are present, gainsmay be severely decreased.The models presented in this essay differ from those in the existing literature in a number ofways. First, the models (of this essay) focus on trade in a unique type of intermediate goods.These intermediate goods are inputs to a production function that allows harvesting (andsubsequent consumption) of a final good whose stock is internationally open access. Second, oneof the models presented in this essay is a two-stage game where the production decision istemporally separated from the trade decision. This allows the nations to manipulate price ratherthan take it as given. Third, differing levels of the open access resource stock - from severelyrestricted to unlimited- are analyzed in the models. The levels markedly influence trade.This essay explores trade in a world with one primary factor2,two intermediate goods, one finalgood, and two nations. The one primary factor, production time (labour), has only one valuableuse - to produce the two (intermediate) goods. These two intermediate goods are the only inputsto a production function that generates a level of harvesting capacity of a stock of aninternationally open access resource. The utility of each nation is a function of the amount of2 The common fishing ground can also be considered a primary factor. However, it does not enter into thealgebra of the model.88the open access (final) good it harvests. Analysis of the model reveals that when the stock of theopen access (final good) resource falls below the combined autarkic harvesting capacity of thenations, no trade occurs. Analysis also reveals that when the stock of the open access (finalgood) resource is above the combined harvesting capacity of the nations were they to trade underthe assumption of unlimited stocks, trade does occur and both nations increase welfare overautarky levels. Further analysis reveals that when the open access resource stock level liesbetween these two states, trade still occurs but with different divisions of the welfare gains overautarky depending on whether the one-stage or two-stage game is implemented.The literatures on the economics of open access3 resources, on the economics of renewableresources, and on trade involving renewable (and non-renewable and exhaustible) and openaccess resources is all related to the present study. Brander and Taylor (1994) provide anexcellent review of the literatures of the limited, renewable resource problem and related tradeeffects. Their paper analyzing interaction of trade and open access renewable resources presentsa strong case against trade when the increased resource exploitation that results from tradecauses renewable stocks to fall below a critical level for sustainability. The resource literaturerelated to inefficiencies arising from open access resources starts with Gordon (1954). Kemp andLong (1984) review the related trade literature, focusing mainly on non-renewable resources.Somewhat recent literature on inefficiencies generated by open access resources include Bolle(1980), Khalatbari (1977), Sinn (1984), Reinganum and Stokey (1985), and Mason and Polasky(1994).Brander and Taylor (1994) explain the important differences between resources that are common property(collectively owned) and open access (unrestricted access). Open access resources create market failures.89Markusen (1976) is the closest in subject to this essay. He considers both the market failurescreated by open access resources and by imperfect competition. He models the two usualexternalities arising from open access resources: the intertemporal and the interdependent4.Theintertemporal externality arises because, in a multi-period analysis, the future stocks of theresource depend on the current stocks. This essay does not consider this intertemporal externalityin the present analysis5 (although it may be considered in future work) in order to simplify themodel and focus directly on the interdependence externality. The interdependence externality issimply the “commons” problem in which one nation decides upon a harvest rate withoutconsidering the effects on the other nations. Markusen’s model is a two primary factor, two finalgood, two nation world where one primary factor is the open access resource. The differencesof Makusen’s model to the current essay lead him to the different conclusions that countriesproducing from an open access resource can influence each other both in terms of availablefuture stocks and willingness to trade6.There is also some literature on resolving these externalities that may be of interest. For example, Samuelsonand Messick (1986) found that experimental subjects will attempt to resolve inefficiencies generated by thecommons externality (and probably the intertemporal one as well). Their subjects often elected a superordinateauthority in order to divide the use of the commons more efficiently.The intertemporal externality is simply assumed away in this paper. The paper only looks at the present timeperiod (only one harvest stage) and ignores the future. Such an approach is reasonable under a number ofalternative assumptions: very high discount rates, very high resource stocks, or some restriction on the harvestsuch as banning any harvesting of stocks under a certain size (e.g., as in fish, where the banned fish - the babiesand propagating adolescents- replenish the stocks to the same level every period with the babies becoming thenext adolescents and the adolescents the next harvested adults).6 One main difference that makes Markusen’s conclusions different is that in his model (and in much otherliterature on this topic) competition for goods coming out of the open access stocks takes place before trade.This allows one nation to specialize in the resource intensive good, in general, thus relieving much of thecompetitive effects. In this essay’s model, competition is forced to be downstream of trade ensuring that the fulleffects of competition are considered in trade.90The literature on the economics of trade with imperfect competition (non-monopoly) starts withBrander and Spencer (1985). They find that nations which can, do attempt to shift economic rentto the domestic economy, for example, by subsidizing certain domestic industries. Thisindividual maximizing behaviour by all nations results in investments made to redistribute rentsrather than maximize the total rents available. The outcome is an inefficient equilibrium. Thetwo-stage model of this essay yields a similar result when the open access resource stock isunlimited.The remainder of this essay is divided into four main sections: The main assumptions, the modeland the autarky case are outlined in the next section. That is followed by the analysis sectionsof the one-stage game and the two-stage game, and then by a summary and conclusions. In theanalysis sections, four cases are examined altogether: the one-stage game without and withrestrictions on the open access goods stock, and the two-stage game without and with restrictionson the open access goods stock. This ordering of these cases provides, respectively, thebenchmark case (of Ricardian-like trade), the effects of the open access goods externality, theeffects of imperfect competition, and the combined effects of both the open access resourceexternality and imperfect competition.2. The Assumptions, the Model and the Autarky CaseIn this section, the major assumptions are formalized in order to construct a model of the91situation. Production functions are presented (and utility functions are discussed) for optimizationin the analysis sections that follow.Consider two nations and a common resource. Nations A and B can produce fishing nets andfishing boats according to their own technological and labour endowments. The populations ofboth nations (which are assumed to be equal) consume only one final good - fish. Although thewater boundaries may be owned by either nation, the stock of uncaught fish (and the fishinggrounds) are a common (open access) resource7.The “final” consumption good - the fish - isassumed to be produced through the use of the intermediate goods, nets and boats, on thecommon fishing ground8.Production time is the only primary factor in production of the intermediate goods. Time hasno other uses of value. Similarly, the intermediate goods have no other value except in their usefor harvesting fish. The wage here is determined by giving all workers equal proportions of thefinal good harvested (that is, if it is assumed that they all worked to their required productivitylevel in either intermediate good). This situation, as described, is basically a one primary factor,two intermediate goods, one final good, two nation world9.The caught fish are not common property. The are owned by whomever catches them.Another example of the common resource of two nations would be an oil field that can be accessed from eithernation. In this case, the shared resource would be non-renewable.This may not be the most general description of the model. In the most general description, each nation has onetime and two productivity endowments (where “time” may represent a bundle of necessary production inputs).The two productivities represent the manufacture of any two goods. Utility is represented by the consumptionof the final good (fish in this case). When the stock of the final good is unlimited then the consumption can berepresented by the production function given the two intermediate goods alone. When the stock is restricted thenrelative fitness matters and the relative harvesting capacity can be interpreted as apotential utility function (i.e.,92Each nation is endowed with H units (e.g., hours) of production time. Nation A can produce r1nets or r2 boats per unit of production time. Nation B can produce r1 nets or r2 boats per unitof production time. It is assumed, without loss of generality, that Nation A has the comparativeadvantage in nets so when the nations trade, A gives up nets in exchange for boats from B.The nations are fully rational and have complete information at each stage in the game.The harvesting capacity of the final good is summarized by a production function based on theassumptions of the model outlined thus far. The most general tractable functional form isassumed for the production function. The function is the log-transformation of the Cobb-Douglasform. The intermediate goods are the inputs. These intermediate goods are combined with theopen access resource (i.e., the common fish stocks) to produce the final good (i.e., caught fish).it represents consumption if the other nation does not exist). Each nation ends up consuming the actual harvestlevel when the other nation does interact. Under these more general descriptions, this model can be consideredin a less specific manner if need be.93Define each nation’s “harvesting capacity” to be K (K = A, B) where:= a Ln[rM (H — xA) — t] + (1 — a) Ln[r XA + p t] (3-1)= a Ln[rlB (H — xB) + t] + (1 — a) Ln[r XB—p t] (32)where: t = quantity of good one exchangedp = price of good two for good one1>a>Owhere__a__) is the weighting measure of good one vs. good two1-awhere a is the technology parameter reflecting the relativeimportance of input one into the production offishXA = time Nation A spends making good twoXB = time Nation B spends making good twoH = amount of production time available for each nation= Nation A s productivity for good one= Nation A productivity for good twor18 = Nation B “s productivity for good oner2B = Nation B’s productivity for good two94Now defme each nation’s fish production function (representing utility) to be Z (K = A, B)where:yZA=Min[YA,( A )S]yZB = Min[YB, B ) sjwhere: S = available fish stocks.Thus, the harvesting capacity function specified above represents the amount of the final gooda nation consumes when fish stocks are unlimited. When fish stocks are limited then a nation’srelative fitness matters (as measured by the ratio of their harvesting capacity to the combinedharvesting capacity of both nations). As well, consumption of fish (utility) will also depend onthe level of available fish stocks.The functional form of the harvesting capacity contains only one primary factor, the productiontime, divided between work on intermediate goods one and two. These are the only two inputsto the production of the harvesting capacity. Each is necessary for harvesting capacity as longas a does not equal zero or one. This form exhibits decreasing returns to scale (DRS) in inputswhen they are increased and the original harvesting capacity value is larger than Ln[2] units10.Constant returns to scale (CRS) occur in inputs occur when they are increased and the original harvestingcapacity is Ln[2] units. The same results for both of these cases (i.e., DRS and CRS) occur in terms of theprimary factor, H, when it is additionally assumed that when H is increased XK (K = A, B) is increasedproportionally (i.e., the proportional division of the time H is unchanged when the increase occurs).95This is assumed to be the case in the rest of the essay.The optimization of any reasonable utility function (one that is either CRS or DRS in this contextto give non-increasing marginal utility) will correspond with that of the harvesting capacityfunction because the utility is a function of the harvesting capacity.Again, the only time that the harvesting capacity does not equal the fish production is when thecombined harvesting capacity of the nations exceeds the final goods stocks. Thus, two levels offinal resource stock will be analyzed in this essay. The first level is labelled unrestricted. At thislevel, stocks are in excess of the combined equilibrium harvesting capacity of the nations whenthose nations are trading under the assumption of unlimited resource stocks. The second levelis labelled restricted. At this level, stocks lie below the combined equilibrium harvesting capacityof the nations when those nations are trading under the assumption of unlimited resource stocks(i.e, free trade levels).96Thus, define the following levels of fish stocks:SL = the level offish stocks depleted under autarkySH = the level offish stocks depleted under free tradewhere: S 5H for unrestricted fish stock levelsSH> S for restricted fish stock levels5H> SLFor the upper range of restricted stocks, S, such that SH > S > SL the nations still have anincentive to trade (to increase their harvesting capacity over the autarkic levels). However, thenations cannot reach the equilibrium trade level they enjoyed when stocks were unrestricted. Therestricted stocks are divided up depending on a nation’s relative fitness, which is determined byits relative harvesting capacity as defined in the fish production function, ZK. The outcomes ofthe games when stocks are restricted differ from those when stocks are unrestricted. Games withrestricted stocks have payoffs (final goods consumption measures) that directly include theharvesting capacity of the other nation(s).Now consider the equilibrium of the model under autarky. Each nation will optimize itsproduction division decision based on the no-trade assumption. The actual amount of availablefish stocks is irrelevant because by optimizing its harvesting capacity, a nation also optimizesits relative fitness when stocks are restricted. Therefore, Nation A will optimize (3-1) withrespect to XA, and Nation B will optimize (3-2) with respect to XB, taking p and t equal to zero.First (and second) order conditions for maxima give the following:97ay—=0 x4=H(1-a)4autarky= a Ln[rM H a] ÷ (1 - a) Ln[r H (1 - a)]—k =o XB =H(1 -a)aXB= a Ln[rlB H a] ÷ (1 - a) Ln[r, H (1 - a)]3. Analysis of the Trade in a Ricardian World - The One Stage GameIn this section, the one-stage game is defined and analyzed under different levels of the openaccess (final) good stock. Nations have no influence over price in a one-stage game and so aRicardian outcome is expected where nations make production and trade decisions based oncomparative advantage.In the one-stage game the nations simultaneously choose their production division (A choosesXA(P), B chooses x(p)) and terms of trade (A and B each offer their own t(p) function) in theone stage. In the trading equilibrium the price is where the t(p) function of Nation A intersectswith that of Nation B. This is equivalent to a Walrasian determination of the equilibrium price.Case i. One Stage Game with Unrestricted StocksThis is the benchmark case. The nations trade only when there exists some comparativeadvantage between them - when r2A/rlA r2B/rlB. The equilibrium trading price will always fall98on or between these two production ratios. However, both nations will not be indifferentbetween trading and not trading only when the price of good two in terms of good one (i.e., theamount of good two bought for one unit of good one) falls between r2A/rlA and r2B/rlB.Two kinds of equilibria are possible for this case: 1) full specialization by both nations; and 2)full specialization by only one nation. This is the standard Ricardian result - that at least onenation fully specializes production in the trading equilibrium. Although this result is acceptedand may be found in most economics (of trade) textbooks like Krugman and Obstfeld’s (1991),it is proven for this particular model in Appendix Fourteen.In order to find the equilibrium production divisions (xA and xB) and terms of trade (p and t),first optimize equations (3-1) and (3-2) for trade level, t. It has been assumed that each nationspecializes its production where it has the comparative advantage. Nation A specializes in goodone, Nation B in good two (as it was assumed that rlA/r2A > r1JIr2previously). Optimizationreveals:Nation A ‘s optimal t = - [(1— a) p rM (H - xA) - a r XA] (33)Nation B’s optimal t = - [—(1 — a) p T1B (H— xB) + a r2B x8] (3-4)In a competitive equilibrium, the market clearing price is such that the two desired trade levelsare equal.99If it is assumed that both nations fully specialize then XA = 0 and XB = H. At these values, theother equilibrium values become:ap= , t=(1—a)r H(l-a)rMSince it is not acceptable to simply assume that both nations fully specialize, the equilibriumproduction division decisions of each nation are determined through optimization.Equations (3-1) and (3-2) are used to optimize for values of XA, x, and t independently (i.e.,Nation A optimizes its choice of XA and t over equation (3-1) and Nation B optimizes its choiceof XB and t over equation (3-2)). From the previous analysis on equations (3-3) and (3-4), theprice is determined from the market clearing condition (that the desired trade levels are equalin equilibrium). The optimization over the production decision for each nation give the followingconditions:= (rlArH—r,t)(l —a) aprt (35)TM T14XB= (riB T2B H + r28 t) (1 — a) + a p TIB t (3-6)T1B T2BAt least one nation will not satisfy its marginal conditions for an interior solution of itsproduction decision at the values assumed, instead attaining its corner solution (i.e., its fullspecialization level with its first derivative increasing in that direction and second order100conditions to verify that the level is near a maximum). For example, when the intermediategoods are equally weighted (when a = 1/2) and each nation has an absolute advantage in onegood then neither nation satisfies its marginal conditions for an interior solution but instead eachnation hits its full specialization corner solution (i.e., XA = 0 and XB H). Thus, theequilibrium values first assumed (i.e., full specialization by both nations) do correspond withthe computed equilibrium under certain conditions.When stocks are unrestricted (with the price, as determined endogenously by equilibriumconditions, falling between comparative advantage ratios), the nations will shift production totheir area of comparative advantage and trade to an efficient level. The general result is that eachnation increases its harvesting capacity over the autarky level, where harvesting capacitycorresponds to consumption (given unlimited final good stocks):Full Specialization 7?’ade YA = a Ln[rM a H] + (1 - a) Ln[r2B a II]Full Specialization ifrade B = a Ln[rlA (1 — a) H] + (1 — a) Ln[r28 (1 — a) H]Full Specialization itade p=a1-a rMFull Specialization 7J’ade t = (1 - a) rM Hwhere, for obtaining Full Specialization 7Jade the following is required:arM( )rlB [fr?]1-ar2B(a)r2.4.r4 awhere [fsl] together with [fs2] gives: —______—rlB 1—a r2B101Combining the two conditions for full specialization by both nations reveals that comparativeadvantage is not the only requirement. First order conditions for full specialization by bothnations not only require that there be comparative advantage but also that where each nation hasits comparative advantage that it must also have sufficiently high relative advantage as well (asthe two conditions above - [fsl] and [fs2] - show11. The first condition must be satisfied forB to fully specialize when A does, the second for A to fully specialize when B does). Thus, ifthe goods are equally weighted and comparative advantage exists, it is still possible that fullspecialization by both nations will not occur in the trading equilibrium. This matches thestandard Ricardian result where at least one nation will specialize if there is comparativeadvantage, but for both nations to fully specialize entails further requirements on relativeproductivities’2.The gains in harvesting capacity for each nation, as measured by the difference in capacitybetween when the two nations are trading under full specialization (by both nations) and whenthe nations are in autarky, is:“ The conditions required for full specialization by both nations in either the one-stage or two-stage game areformulated in the same way through first order conditions. In order for A to specialize fully given that B alreadyis requires that: OYA/8X 0 at XA=O and x5=H. Similarly, for B to specialize fully given A already isrequires that: aYB/8XB 0 at XAO and X5=H. The partial derivatives are evaluated using optimal p and tvalues as defined by (3-3), (3-4), or p’, and t from the two-stage game analysis that follows; and knowingthat ru > 0, 1 > a > 0 , H > 0; and checking that second order conditions are satisfied for maxima.12 See Appendix Fourteen for proof that it is the case that at least one nation fully specializes when comparativeadvantage exists (and both nations want to trade).102A YA =(1 -a)(Ln[r2BaI-I] -Ln[r,(1 -a)H])>OA = a (Ln[rlA (1 - a) H]- Ln[rlB a H]) > 0under the conditions speced aboveRe-arranging, the condition for the absolute gains from trade of nation B to be greater than thoseof nation A is:a Ln[r H (1 - a)] + (1 - a) Ln[r (1 - a) H] >a Ln[rlB a H] + (1 — a) Ln[r2B a H]If a = 1/2, so that the goods are equally weighted, then the left hand side of the inequalitycorresponds to A’s autarlcic harvesting capacity and the right hand side to B’s. The initiallyinferior nation (B in this case) would obtain a greater (relative and absolute) increase inharvesting capacity when trading occurs. This is not always the case, but the initially superiornation can obtain the greater gain only when the weighting (a) favours its comparativeadvantage.One nation is indifferent to trade when conditions are such that only one nation fully specializesin equilibrium. In this case the nation that is not fully specializing obtains its autarky level offinal goods whether it trades or not (assuming that side payments are ruled out), and so maychoose not to trade. The equilibrium price in the case of only one nation fully specializing is thenon-fully-specializing nation’s internal price of the intermediate goods (i.e., the slope of itsproduction possibilities frontier). This makes that nation indifferent to costless international tradebecause it can achieve the same result internally. For example, A does not fully specialize under103the assumption that B does, when:1 -aT2B = ( ) r K where 1 > x > 0aEquilibrium for this condition, assuming that B does fully specialize at the equilibriumconditions, gives:rxA=H(l —a)(1 —K), p=—, t=(1 —a)HrMKTL4Y(xA*, p. t) = Autarky 1’4Nation A only obtains its autarky level of fish under these conditions. Since such a nation wouldbe indifferent between carrying out the trade that only benefits the other nation, it may wellchoose not to trade13. This result, again, corresponds with the standard Ricardian result of whatoccurs when only one nation fully specializes in equilibrium.Only trade under bi-nation full specialization for the one-stage game will be considered in theanalysis that follows (the other case of unilateral full specialization with trade is considered aspecial case and left for future work). When both nations fully specialize, both nations obtaingains from trade.Comparative statics are used to determine the effects of certain parameter changes on bi-nation13 However, this indifference can be broken in favour of trade by the nation who benefits more when the tradeoccurs. Side payments are not possible in a one-stage world; but, the enticing nation (which wants to trade)could soften its terms of trade offered in order to break the other nation’s indifference. Simply by asking forepsilon less in exchange, the indifference can be broken and both nations can be better off (with the enticingnation taking most of the gains from trade).104full specialization equilibrium harvesting capacity, OY, and on gains from trade, SAY. The gainsfrom trade are measured by the change in a nation’s harvesting capacity over autarky (whereharvesting capacity equals consumption when final stocks are unrestricted). These comparativestatics correspond with intuition in general:Partial Derivative aYA aAYA ÔYB aAYB/arlA + 0 + +/ôr2A 0 - 0 0/arlB 0 0 0 -/ar2B + + + 0/aa + :10w a + :low a -:high a + :low asign[r1-r2] ?:mid a sign[r1-r2] ?:mid afor other a -:high a for other a -:high asee (csl) see (cs2)‘lable ‘Iwo: Comparative Statics of the One-Stage (lame with Unrestricted Stocks1058AY 1A+ Ln[r (1 — a) H]— Ln[r2B a (Cs 1)a8AY 1B= —( ) + Ln[r (1 — a) H] — Ln[rlB a II] (cs2)1-aWhen a nation increases its productivity where it has a comparative advantage (i.e., A in goodone making r14 increase, or B in good two making r2B increase) only the other nation obtainsan increase in its gains from trade (although both increase harvesting capacity over the levelsbefore the productivity increase) in general. The nation that experiences the increasedproductivity cannot specialize further if it is already fully specialized so it will not obtain anyincrease in harvesting capacity relative to autarky capacity. The other nation does obtain anincrease in its gains from trade as it now gets more of the other good in exchange for its own,as price adjusts in its favour. This adjustment occurs because price is based on the relativeabundance of the two goods. The adjustment does not favour the nation that experiences aproductivity gain (where that nation is currently specializing) as the relative abundance of thatgood would increase making it worth less.When a nation increases its productivity where it does not have a comparative advantage (i.e.,A in good two making r2 increase, or B in good one making rlB increase) then only that nationhas a decrease in its gains from trade (although neither nation changes harvesting capacity overthe levels before the productivity increase) in general. The nation that experiences the increased106productivity does not change its specialization if the increase does not change the comparativeadvantage between the nations (no change is assumed). Price and production under trade isunaffected so the nation that does not experience the gain in productivity has no change to itsgains from trade. However, the nation that does experience the gain in productivity increasesits autarkic harvesting capacity and so decreases its gains from trade as a result.When a is at extreme values (high or low), there is not room for much further specialization byat least one nation as that nation moves from autarky production division to full specializationunder trade. As a result, the gains from trade (when nations fully specialize under trade)decrease as the weighting factor (of production inputs not consumer tastes14) moves towardmore extreme values. Thus, decreasing a further when it is already small decreases the gainsfrom trade, as does increasing a more when it is already large. When a is in the intermediatevalues the other parameters determine whether either nation increases its gains from trade. Thenation with the comparative advantage in the good that is weighted by a always obtains anincrease in autarky welfare (and the opposite for the other nation) when a increases. However,both nations either increase or decrease their consumption level under trade depending onwhether the specialization productivity in the good weighted by a, rlA, is greater or less than thatweighted by (1 - a), r2B. Regardless of whether there is an increase in welfare under trading(when both nations specialize fully), a nation’s gains from trade do not necessarily alwaysincrease as well.14 Changes in a affecting welfare and gains from trade are worth noting. This weighting factor could change astechnology changes the input requirements of the production function. If a is considered this way, and notsimply as a taste parameter, the results may be more meaningful.107Overall, the analysis of the one-stage game with unrestricted stocks matches the standardRicardian results found in most economic (trade) texts, such as Krugman and Obstfeld’s (1991).Nations are price-takers and, in equilibrium, at least one fully specializes when nations trade toeach one’s benefit.Case ii. One Stage Game with Restricted StocksWhen the resource stock is restricted to a level at or below the combined autarkic harvestingcapacity of the nations, SL, then the nations do not trade at any price.At these low stock levels, at least one nation will not want to trade because the game is zero-sum. Since, in this case, trade cannot increase the total size of the catch but can only alter theshares going to each country, there is no opportunity for mutual gains from trade. There is onlyone price (per trade amount considered) that leaves the relative harvesting capacity unchanged(from autarky levels) under trade. However, this is not the equilibrium price (based on eitherRicardian or autarky production) in general because the conditions for a price that does not alterthe relative gains from trade does not correspond with the conditions for an equilibrium price15except in pathological cases. Thus, assume that the pathological case does not hold. Therefore,trade will not take place when the level of common stock is at or below SL.15 The equilibrium price is determined by the intersection of the optimized t(p) offer curves of the two nationswhere optimization is through the first order condition. In the function assumed, this gives an equilibrium pricewhich is determined without having to evaluate any natural logarithms. The price that maintains the relativeharvesting capacity of the two nations when they trade is determined by the relative harvesting capacities. Theseinvolve evaluating natural logarithms. It is therefore reasonable to assume that the equilibrium price will notusually equal the relative-harvesting-capacity-maintained price.108When the common resource stock is above SL then an equilibrium with trading can exist. As inthe case of unrestricted stocks, the nations trade only when the trading price of the intermediategoods lies between their comparative advantage ratios.Equations (3-5) and (3-6) (or their corner solution counterparts) still have to be satisfied in orderto give the optimal production division for each nation. However, now it is not possible toensure that equations (3-3) and (3-4) are satisfied. Nevertheless, they will still want to trade asgains from trade are available to be split.A new requirement defining the trading level is introduced when the fish stocks are resthcted.In equilibrium there is assumed to be no excess harvesting capacity. Thus, in equilibrium, thecombined harvesting capacity will equal available fish stocks. Nations will not trade to an excessof harvesting capacity because by doing so the relative fitness of at least one nation will decline.The nations also will not trade to a shortage of harvesting capacity because each would still standto gain if they traded more. This equilibrium condition defines the terms of trade:Equilibrium (t, p) is such that: YA + = (restricted fish stock Zevel)where:‘A defined by (3-1) 1’B defined by (3-2)Solving the system requires terms of trade, (t, p), such that demand, as characterized by thetotal available harvesting capacity of the fish stocks, equals supply, as characterized by theavailable open access fish stocks:10917A + = S at (t*, .p*)where : XA of YA = Max[0, XA defined by (3-5)]XB of‘B = MinIH, XB6 defined by (3-6)1*= __aT24 XA ÷ T2B XB )1— a rM (H — XA) + TIB (H — xB)5H> S> SLA simple numerical example may be helpful. Assume that r1= 2, r24= 1, r,,= 1, r2B= 3/2,a = 1/2, H 10, and S = 3.887 units. These values give the following equilibrium solution:xA=3.95, xB=6.22, p = 0.837, and t = 1.568. Nation A gains 0.051 units of harvest over itsautarky level while nation B gains 0.067 units over its autarky level.Equilibrium specialization when fish stocks are restricted will never be greater than (and willusually be less than) the equilibrium specialization when fish stocks are unresthcted because theamount traded in equilibrium will always be less.Also, in equilibrium with restricted stocks, neither (3-3) nor (3-4) is satisfied in general’6.However, each nation does gain in harvesting capacity over its autarky state. The inferior nationin autarky, Nation B, may gain a larger relative and absolute amount in equilibrium (as it doesin the numerical example above).16 Strictly speaking an altered form of (3-3) and (3-4) would continue to hold. This altered form would recognizethe discontinuity implied by the exhaustion of the available fish stocks (and the potential for further gains fromtrade).110When S is restricted to a level between SL and S11, then the equilibrium of the one-shot gameinvolves: production specialization not greater than (and usually less than) free trade levels; tradeto a level that is efficient for the resource stocks available; and an increase in harvest to eachnation from autarky levels.When resource stocks are restricted to a level at or below the combined autarkic harvestingcapacity, then nations do not trade. Each nation divides its production to maximize its autarkycapacity (i.e., its fitness level in this case). Fish stocks (and utility) are then distributed basedon each nation’s fish production function (i.e., ZA and ZB). The price taken is irrelevant becauseno trade occurs, and harvesting amounts decrease as the fish stock level decreases.4. Analysis of the Trade with Production Precommitments - The Two Stage GameIn the two-stage game, the nations simultaneously choose their production division (A choosesXA, B chooses xB) in the first stage, and their terms of trade (each nation offers its own t(p)function) in the second stage (see Figure Five for timing)17. The nations have the rationalityto work backwards from the second stage to the first to optimize their production decisionsknowing how that will affect the amount that can be traded and so the trading price (which isdetermined by the Walrasian auctioneer) as well. The method of determining the trading price17 The sequential decision making of the two-stage game may reflect reality better than the one-stage game for thefollowing reasons: There is less possibility of reaching a non-trading equilibrium in the two-stage game asnations would always attempt to trade after the first stage, if it were possible, regardless of their previous beliefsabout trading. Also, the production processes of the nations may be sequential and not simultaneous.111is known beforehand. It is assumed to be the Walrasian (market-clearing) price based on therelative abundance of the two intermediate goods after production of these goods is complete18.This allows the nations to affect price through their production decisions - their production precommitments (i.e., they do not take price as given as they did in the one-stage game).Production and Trade Time Linefor the Two Stage Game:Intermediate goods Final goodsproduction occurs. production occurs.Trade occurs.0 1 2I ITimeProduction Plan Ex-post agreement Finalset for each occurs. consumptionnation’s occurs.intermediategoods.Figure Five: Time Line of the Two Stage GameThis Wairasian market clearing solution also corresponds to a Nash Bargaining Solution that uses threat pointsbased on end-of-first-period states. Nations bargain over a harvesting capacity proxy in the solution. SeeAppendix Twelve for details.112Case iii. Two Stage Game with Unrestricted StocksIn comparison to the one-stage game, the nations will always attempt to manipulate the tradingprice of the intermediate goods to their own advantage. This is possible in the two-stage gamebecause the terms of trade allow influence through previous actions (those which occur in theprevious stage). It is not surprising that this two-stage process, where nations alter theirbehaviour in an attempt to influence price, will have different equilibrium outcomes than theone-stage game where the nations were price-takers.Now consider the case where the nations are in stage two, having made their productiondecisions, and are optimizing for their own trading levels. Nation A would optimize equation(3-1) for t in terms of p and the other parameters, and talcing XA as given. Nation B wouldoptimize equation (3-2) for t in terms of p and the other parameters, taking XB as given. The twot(p) schedules would then be used by the market (a Walrasian auctioneer) to find the commont (and the equilibrium p). Solving for the equilibrium p this way gives:a T X+T X___it ) (3..7)1—a rM(HxA)+rlB(HxB)The equilibrium p is simply the ratio of the amount of good two available to the amount of goodone available (where good one and good two are the intermediate goods). The amounts aremeasured at the beginning of stage two and weight-adjusted to reflect the relative importance ofthe two goods. (For example, if good one were much more important than good two asexpressed by the weighting- a high a value - then price would go up so that a nation exchanging113good one for good two would get a lot more good two in return.)The equilibrium trade level, t, can be defined in terms of the other variables by back-substitutingfor p in the optimal t equations formulated from either (3-1) or (3-2). The resulting equilibriumtrade level is:t * * = (1 — a) IA XB (H — xA) — T1B r XA (H — X) (3-8)T X + T XBThe equilibrium t decreases as the importance of the good two (i.e., the good traded for by A)decreases. The importance of good two is measured both by its weighting factor, (1 - a), andby the reciprocal of its relative abundance at the beginning of stage two (where the relativeabundance is r24 XA + rZB x).With all stage-two equilibrium variables defined in terms of stage-one variables, nations canconsider how to optimize their stage-one (i.e., production division) decisions. Both t and p canbe substituted into (3-1) and (3-2) to eliminate all choice variables but XA and XB. Each nationoptimizes its production decision knowing the exchange amount and rate that will occur. Theequilibrium corresponds to mutual best responses (satisfying each nation’s optimal trade andproduction decisions)19.It should be noted that the analyses of Cases i. and iii. do not require the open access resource assumptions.The harvesting capacity functions (3-1) and (3-2) can be considered as utility functions directly. The analysesof Cases i. and iii. give results for a one primary factor, two final good, two nation world where utility is afunction of the two final traded goods directly.114Equation (3-1) is maximized over XA while (3-2) is maximized over XB (given that the p andt” definitions have already been substituted into these equations). The maximizations result ina system of two equations (the first order conditions for (3-1) and (3-2)) in two unknowns (xAand xB). The solution of the system is (xA**, xc). The second order conditions at this solutionare used to verify that the maximum has been found20.Unfortunately, the system of equations does not have a simple, easily interpretable closed formsolution. The system is a set of two “cubic” (i.e., fourth order) equations21.One of the cuberoots to each of the equations gives the correct equilibrium productivity decision for each nation.This seemingly simple system cannot be solved algebraically in general22.Therefore, a representative numerical example is focused on for the analysis that follows (i.e.,the example represents a comprehensive testing of the parameter space using computersimulations).20 The solution must also meet the additional requirement that the production division is feasible.(i.e., that: H XA** 0, H XB** 0, and 1t < r XA + rm xB).21 These fourth order equations are of the form:CIXA ÷ C2XB ÷ CSXA2 + C4XAXB + CSXB2 + C6XA3 + C7XA2B ÷ C8XAXB2 + C9XB = 0d1 XA + d2xB + dsxA2 ÷ d4xAxB + dSxB2 ÷ d6xA3 + d7xA2B ÷ dgxAxB2 + d9xB3 = 0where c1 and d1 are determined by the productivity, weighting, and production time parameters.The problem in solving the system analytically arises because there are non-real (i.e., imaginary) parts to dealwith in cubic roots that are left in algebraic form. These imaginary parts do not easily reduce in general.Furthermore, solving the system itself translates to solving an equation of greater than fourth order, and noclosed form algebraic solutions exist for such equations.115Consider again the following example:H=1023rM=2 r=1 T1B=l r=—The solution to the system of equations for this case is:** = 5 (139189560 - 57995650 /) 3.6302A 53355998= 5 (—38 ÷ 58 6.7890B 46t 4.2739 0.8660The second order conditions are satisfied at these values.The resulting harvesting capacities at this equilibrium solution are larger than those under theautarkic conditions:Autarky YA 1.956 Equilibrium‘A 2.064Autarky B 1.8 12 Equilibrium B 1.941It can be seen that the equilibrium solution does not correspond to the most efficient division ofproduction among nations. The nations do not do as well, generally, under this two-stage gameas under the one-stage game where the Ricardian outcome results. When nations trade underconditions of imperfect competition, inefficiencies may arise as Brander and Spencer (1985) haveshown.When nations believe that they can alter price they will make (production) decisions to obtain116the best terms of trade for themselves. By attempting to maximize their own welfare in this waythey may not maximize the joint welfare. When the equilibrium of individual maximization (ofharvesting capacity) in the two-stage game does not correspond with that of the one-stage game,potential Pareto improvements are available. Movement towards these improvements can beconsidered to be a Prisoners’ Dilemma. If both nations defect they remain at the two-stage gameequilibrium. If both nations cooperate they can move towards the one-stage game equilibriumand both take welfare gains (i.e., gains in harvesting capacity)23.The Ricardian outcome and joint-maximization outcome can both result in much greater welfaregains from trade (as measured by the number of fish caught) in general (the Ricardian exampleis shown in Case i.).The two-stage equilibrium can result in full specialization as in the one-stage game, but theconditions required are more restrictive than those of the one-stage game. First order equationsare used to define the conditions, and second order equations to ensure that the full specializationequilibrium is the maximum corner solution. For nation A to fully specialize while B is fullyspecialized requires:If one nation cooperates while the other defects then the cooperating nation loses harvesting capacity while thedefecting nation gains. The cooperating nation loses because it specializes more, which benefits both nations,but is no longer at its best response point to the defecting nation, and by definition must experience a loss inwelfare. The defecting nation gains although it is not at its best response to the new position to which thecooperating nation has moved. The defecting nation benefits through the increased specialization of thecooperating nation regardless because the increased specialization is a (strategic) complement to any level ofspecialization of the defecting nation. The result is a Prisoners’ Dilemma where the dominant strategy is toremain at the two-stage game equilibrium point, although that is not jointly efficient.1171- a22)raand for nation B to fully specialize while A is fully specialized requires:a (2 - a)r r(1 - a)2These restrictions require that the nation producing a certain good needs to have a higher relativeadvantage in that good than under the one-stage game to allow that nation to fully specialize inthat good in the trading equilibrium. The stronger conditions on the two-stage game over theone-stage game in this example of full specialization by both nations in the trading equilibriumimplies an apparently general conclusion for any trading equilibrium: For a given set ofparameters (productivities, weighting, and productivity time allotment), the trading equilibriumof the two-stage game will always involve less specialization than the one-stage game unless itis the case that both equilibria allow full specialization by the nations. This conclusion has beenreached based on extensive numerical analysis of the two-stage game equilibria24.When the Nash equilibrium does not align with the jointly optimal outcome, a policy-makerwould be interested in finding how to change some conditions to improve efficiency. For24 As well, a mathematical proof for a special case can be found in Appendix Fifteen to support this hypothesis.118example, consider the two-stage game if the first stage had a Stackelberg leader25.An increasein harvesting capacity for both nations results with a Stackelberg leader when the equilibriumharvesting capacity of the one-stage game is larger than that of the non-Stackelberg two-stagegame. Although both nations gain, the Stackelberg follower gains more than the leader, asmeasured from the simultaneous Nash equilibrium. This occurs because the increases inspecialization are strategic complements (see Appendix Thirteen for details).There axe other ways to alter the equilibrium besides changing the rules of the game (e.g., bymaking one firm a Stackelberg leader). To explore these other ways, it is useful to know howchanges in key parameters affect the outcome of the trade. Comparative statics provides someinsight.Modelling the game as Stackelberg may be appropriate in some cases. For example, if it is overlappinggenerations who are trading instead of nations, it may be the case that the older generation will have to lead.119Comparative statics align with intuition generally26:Partial Derivative aAYA ÔYB/3rlA + + + +/ar2A + - - -/3rlB-- + -/3r2B + + + +13a -:low a + :low a -:low a + :low a+:mida ?:mida -:mida ?:mida+ :high a -:high a + :high a -:high aTable Three: Comparative Statics of the Two-Stage Game with Unrestricted StocksWhen Nation A increases its comparative advantage by increasing its productivity in good one(i.e., rIA increases) then both nations obtain a welfare-increase under trade as well as positivegainsfrom trade. This coincides with the intuition that such an increase in efficiency allows bothnations to shift production further towards where they add most value. The same effect occurswhen Nation B increases its comparative advantage (by increasing productivity in good two26 The analysis that follows is based solely on numerical analysis, as analytical solutions were intractable. Thecomputer simulations so far completed covered a wide range of parameter values, and so there is someconfidence that the comparative statics shown are representative of the general case.120making r2 increase). These productivity increases may benefit the nation experiencing theincrease in productivity in the two-stage game where they do not in the one-stage game, in termsof gains from trade, because they allow both nations in the two-stage game to more fullyspecialize production (where the nations do not specialize further in the one-stage game as theyare already assumed to be fully specialized).When Nation A increases its productivity so as to close the comparative advantage gap withNation B (i.e., r2A increases) then it decreases its own gains from trade as well as those ofNation B. However, Nation A does increase its own welfare in general (i.e., unless it is alreadyfully specializing) because it produces intermediate good two more efficiently. Nation A’s moreefficient production where B adds value (i.e., in good two) increases A’s price elasticity for t(see Figure Six). All else being equal, higher prices for each trade amount increase A’s welfareand decrease B’s welfare, and consequently B’s gains from trade. The same effect occurs whenNation B closes the comparative advantage gap with A through B’s more efficient productionof good one (i.e., when rlB increases). The finding is in agreement with the intuition that closingthe comparative advantage gap (the reason for trade here) will adversely affect at least onenation (holding all else constant)27.The finding runs against an intuition that any increase inproductivity of one nation will benefit all nations in a trading world. The nation experiencingthe productivity increase decreases its welfare gains (over autarky) for two reasons: it specializes27 Consider the case where the nations are “e-close” in comparative advantage terms before the change inproductivity. If nations trade then they experience a wealth greater than that in autarky. Now consider whatoccurs when one nation gets the productivity increase that eliminates the comparative advantage. Now the nationwith the increased productivity will still have a wealth greater than that of the pre-change autarky. However,no trade will occur. The other nation will experience a decrease in wealth over the pre-change (trading) caseas it then reverts to its autarky wealth.121less in response which decreases the other nation’s specialization (as specialization is a strategiccomplement in the two-stage game) resulting in less trade and less gains from trade overautarlcy; and the productivity gain increases its autarky welfare thus decreasing further any gainsfrom trade. The two-stage game outcome differs from that of the one-stage game where thenation not experiencing such a productivity increase incurs no change in gains from trade. Thisis because it was assumed that nations do not change production decisions as a result of theincrease whereas they do in the two-stage game.The preceding analysis has an intuitive implication for government policy regarding where todirect R&D subsidies. Subsidies that increase the production efficiency in any area will increasea nation’s welfare (under trade or autarky) in a two-stage game model. However, only subsidiesdirected at increasing productivity where a nation enjoys comparative advantage will increasea nation’s gains from trade when nations trade under a two-stage game model28. Furthermore,only subsidies in this area are non-detrimental to the other nation, and so will not initiate(welfare-decreasing) retaliatory action29.28 Analysis of comparative statics in the one-stage yields a similar recommendation- direct R&D funding wherethe nation has a comparative advantage in order to increase welfare. Investing in the productivity may also bebetter than investing in ways to make the final good harvesting capacity technology mix be more in a nation’sfavour (i.e., Nation A investing so that a increases). The regions were such an investment is favourable isrestricted by parameters that are outside a nation’s control.29 It should be noted that these recommendations for R&D subsidy direction are opposite what they would be whenstocks are restricted. When nations are in direct competition with each other on how to “divide the pie”, eachwould want to increase its own relative fitness at the detriment of the other nation. To do so entailsstrengthening a nation’s autarky harvesting capacity and this translates into subsidizing R&D in the area ofcompetitive dis-advantage. However, a nation would not want to go so far as to decrease the size of the pie byaltering fitness to the point where there are excess resources that go unharvested (i.e., when SH > S > SL).122Figure Six: Effects of Unilateral Specialization on ElasticityThe only parameter left to comment on is a. Although it is reasonable to expect that theproductivities of the nations will change over time, it may be less reasonable to expect that thetechnology mix that creates harvesting capacity will change over time. If it does then thefollowing analysis is of use. The changes in gains from trade with respect to changes in theweighting factor of the production inputs, a, correspond to those of the one-stage game ingeneral. When a is at extreme values (high or low), there is not room for much furtherspecialization by at least one nation. As a result, the gains from trade when it shifts productionfurther decrease as the weighting factor moves to more extreme values. Thus, decreasing aA’s t(p)PThe Effect of Increasing Unilateral Specialization onThe Price Elasticity of Trade:Nation B’s price elasticityof trade decreases as itincreases its specializationunilaterally.Nation B’s t(p) atgreater specializationNation B’s t(p)t (trade offered at price P)Where A’s t(p) is defined by (3-3), B’s is defined by (3-4), and price is “taken”.The changes in elasticity due to changing unilateral specialization shown are meaningful,but the equilibrium P and t, and the areas bounded by the lines are meaningless here.The gains from trade are not defined by these areas but instead by other equations.123further when it is already small decreases the gains from trade, as does increasing a more whenit is already large. However, at these extreme values, the harvesting capacity under tradeincreases for both nations as the values become more extreme. This is because the gains incapacity from the weighting that increasingly focuses on one intermediate good more thancompensates for the losses in capacity from the decreased trade arising from decreaseddifferences in production specialization amongst nations. When a is in the intermediate values,the other parameters determine whether either nation increases its gains from trade, as is thecase in the one-stage game. However, in general, the nation with the comparative advantage inthe good that is weighted by a obtains an increase in welfare (and the opposite for the othernation) when a increases.As it is noted that in many cases the trading equilibrium of the one-stage and the two-stagegames differ, it is not surprising that the division of welfare gains and rate of change of welfaregains with respect to some key parameters also differs across the games. It may even be the casethat one nation may obtain a greater harvesting capacity in the two-stage game than in the onestage game although it is never the case that both nations would do so at once (see AppendixSixteen for a proof). Therefore, without the possibility of side-payments in the one-stage game,the individual-welfare-improving strategic trade policy of one nation’s government is to attemptto implement the two-stage game construction instead of the one-stage construction.124Case iv. Two Stage Game with Restricted StocksOnce again, the two-stage game outcomes can differ from the one-stage game outcomes whenthe nations trade with S < S11. The difference is again due to each nation’s attempt to influenceprice without regard to its effects on the other nation’s welfare in the two-stage game.The outcome of the two-stage game is the same as the outcome of the one-stage game when fishstocks, S, are restricted to a level (equal to or) below SL. No trade takes place when only twonations are involved. If nations were to trade at least one of the nations would decrease itsrelative fitness, as relative fitness is affected by trade in general. Since a decrease in relativefitness translates into a decrease in welfare in this now zero-sum game, at least one nation willnot have an incentive to trade when excess harvesting capacity already exists in the two nationworld. As a result, no trade will occur30.When S > SL, nations trade and the individual influence of each nation begins to show. Theresults of these influences generate the differences in outcome between the one and two-stagegames.30 However, when more than two nations are involved coalitions may trade (in either the one or two-stage game).The case of coalition trades in the two-stage game when stocks are severely restricted provides an example.When relative fitness is a proxy for harvesting capacity, less fit nations may find mutual trade an effectivemeans of increasing their own harvesting capacities at the expense of a more fit nation. The less fit nations forma trading coalition (or trading block) that excludes the more fit nation. The more fit nation is not be able to alignitself with any other nation as its gains from trading in any coalition is not great enough to form a crediblecommitment to completing a trading agreement with any potential partner. The net result is that the less fitnations increase their harvesting capacities through trading amongst themselves to the detriment of the nationsoutside their coalition.125The equilibria that are analyzed below are considered (as before in the previous sections) to beNash equilibria - they are best responses to best responses. Therefore, the nations obtain theirhighest welfare in these equilibria based on the responses of the other nation. As a result, all theequilibria studied below meet the same restrictions for best responses: XBe XB, XA XA,= A + B, te t, and pe = p (where superscript e are equilibrium values, andsuperscript ** are equilibrium values assuming S = S11).The analysis that follows is based on numerical (computer simulation) study only. The numericalexample previously described in Case iii. will be used as to represent the outcome of severalcomputer runs through many areas of the parameter space.The amount of fish stocks, S, analyzed next can vary between SL and S. In the numericalexample, SL = 3.768 and Sn = 4.005 units of fish. The space between these levels is dividedinto three regions for analysis:1. Region One:In this first region, nations have a choice of producing to autarky levels and then trading to alevel less than or equal to C, or producing to levels which satisfy trading at C. The resultingprice and division of fish will differ between the two choices in general because they aregenerated from two different sets of restrictions. This implies that one nation will be worse offby moving away from autarky production. Thus, it will choose to remain at autarky production.126This forces the other nation to remain at autarky production as well because a move to unilateralspecialization decreases its elasticity and thus its welfare. The choice of production, therefore,is forced to be autarkic in this region.Fish stocks in this region are bounded below by SL and above by the highest level of combinedharvesting capacity that can be attained through trade given that the nations producedintermediate goods under autarky (i.e., they trade in stage two given they made productiondecisions based on autarky in stage one). This upper bound will be identified as S1:aHR1 aHR2S1=a(Ln[ 1)r + r28 r,A + T2B(l-a)HR1 (1-a)HR2+(1-a)(Ln[ ]+Ln[ 1)rM + r18 TM + rlBwhere: R1= rM r, + T1B (1 — a) + a TM rR2 =T1BT2B 4-rMr2B(l —a) +arlBr4Since both nations are producing to autarky levels, the price is easily determined as:r +r r r=2A 2Bwhere: — <p’ <r + T1B TM rlBwhich fails between the internal prices of the two nations.The trade amount is determined by the fish stock level, S, given the known price andintermediate good levels. The trade amount is just enough to ensure that the post-trade combined127harvesting capacity just equals the available fish stocks3t. (More trade would be detrimental toone nation under the relative fitness function assumption. Less trade would leave obtainable extrafish stocks wasted.)The highest amount that can be traded in this Region is restricted by the efficient trading levelas defined by t. In the numerical example, this occurs when t = 2 at a price of p = 5/6. Atthis point Nation A’s fish harvest is 1.988, Nation B’s is 1.855, and combined fish productionis 3.843 units. In this region, it is Nation B, the nation with the lower autarkic harvestingcapacity, that forces the production to autarkic levels.2. Region Two:In this second region, the nations again have a choice on how to produce and trade. They canboth specialize and trade to the efficient level for the fish stocks available or only one canspecialize and let trade obtain the efficient level. The nation that can remain unspecialized doesbetter under the latter choice. Thus, it chooses the production and trade structure in this region.The other nation has an incentive to specialize unilaterally nonetheless as it gains enough fromits division of the increased fish stocks that are available when it specializes to more than offset31 As in Case ii. the equilibrium is efficient in the sense that no resources are wasted (i.e., all the open accessresource stock is used). This efficient equilibrium occurs in spite of the interdependence externality and relatedmarket failure generated by the existence of the open access resource stock. So, although there is a non-pricedeffect and social marginal benefits do not equal social marginal costs at the equilibrium, it is efficient (in animportant sense) nonetheless.128its losses from its decreased elasticity arising from its unilateral specialization32.Only a nation with a higher autarky harvesting capacity can remain at its autarky productionlevel in this region. Increasing its own specialization in this region results in losses fromdecreased elasticity that outweigh its gains from its division of any available fish stocks. Thereason for this asymmetry between the non-identical nations can be found in the Region Oneprice equation. In Region Two, where the nations are close to autarky productions ofintermediate goods, the price is in the favour of the nation with the lower autarkic harvestingcapacity. At this price, only the favoured nation can compensate for losses due to unilateralspecialization through an increase in fish stocks whose division is a function of the trading price.In this region, fish stocks are bounded below by S1 and above by the maximum combinedharvesting capacity attained when the nations trade but one nation remains at autarkicproduction. This upper bound will be identified as S. At S, price will be p, trade level willcorrespond both with t and the efficient level to make VA + B = S, and one nation’sproduction decision will correspond to its optimum, x = x, while the other nation will remainat its autarkic production level. It is useful to separate this region from where both nationsremain at autarkic production and from where neither nation does, as this region differs inequilibrium decisions made by the nations.32 When nations have identical autarky harvesting capacities, this region disappears as neither nation is able toremain at its autarky level and have the other nation experience an incentive to unilaterally specialize.129In the numerical example, Nation B will increase its specialization in production by changingXB from 5 to any level up to 6.164 (which corresponds to XB at these conditions). At this point,which signifies the upper bound of Region Two, the combined harvesting capacity is 3.892 units.3. Region Three:In this third region, both nations specialize, but only one specializes to its optimal level, xr.The nations will again have a choice again between different production and trade methods toobtain the efficient level of trade in this region so that combined harvesting capacity equalsavailable fish stocks. As in the other regions, one nation will be able to “force” a choice thatis better for it while the other nation will have an incentive for accepting the choice and fulfillingthe trade requirements (while it obtains less welfare than under the other choice nonetheless).In this region, as in the second region, it is the nation with the higher autarky harvestingcapacity that forces the choice. Under this choice, the other nation produces to its optimal level,x.J**, while the forcing nation remains at a specialization less than that called for by its XK** level.This allows it a better elasticity (and consequently welfare) than under the other choice. Theother nation has no incentive to deviate from its own (optimal) production level so there is noneed to analyze the incentives of that nation under this equilibrium.In this region, fish stocks are bounded below by S, and above by Sil. The equilibrium in thisregion is characterized by: price equalling p, trade equalling C’ and satisfying the condition130that S= A + Y; and x = for (at least) one nation.In this Region both nations specialize (to some extent) in production where each has itscomparative advantage. Just as in Region Two, in the numerical example, Nation A has theadvantage.Once the upper bound to Region Three has been reached neither nation has the incentive tospecialize further. The gains from being able to obtain excess fish (if there were any) throughfurther unilateral specialization are more than offset by the losses resulting from a decreasedelasticity. (As was noted in Case iii., however, Stackelberg leadership does result in harvestingcapacity gains.)The analysis of the two-stage game under the three Regions of restricted stocks showsdifferences to the analysis of the one-stage game under similar restrictions. A notable differenceis that the stronger nation (in autarky, Nation A in the example,) can choose the form ofproduction and trade structure in most regions of the two-stage game. In effect, the nation withthe higher autarky capacity can influence the division of open access resource stocks to its ownadvantage. It cannot do this in the one-stage game because nations do not act as to affect priceas they are considered price-takers.As a numerical example, consider the case when fish stocks are restricted to a level of 3.887units, like in the example in Case ii. In the two-stage game, nation A gains 0.059 units of fish131while B gains 0.060 units. In contrast to the one-stage game, nation A obtains a larger divisionof the gains from trade here33. Therefore, when the stocks are restricted, the government ofthe nation with the greater autarky wealth may choose to intervene and attempt to controlproduction and trade so as to structure trade between the two nations as a two-stage game.5. Summary and ConclusionsThis essay has analyzed trade when nations can exchange intermediate goods that allow themaccess to a final open access good. In a one-stage game when the open access resource stocksare not restricted, the nations are price-takers and a Ricardian world is modelled where veryefficient (at or near jointly-optimal) trading results. When that resource is restricted to a levelbelow that of combined autarkic harvesting capacity, no trade takes place (in a two nationworld). When the resource level falls between these two points, trade occurs so that no resourcesare wasted. When trade occurs, both nations benefit (i.e., increase welfare over autarky).See Appendix Sixteen for a proposition and proof on this matter. It does show that it can be the case that onenation may find it attractive to implement the two-stage game rather than the one-stage game although this isjointly inefficient.‘ When stocks are restricted the “size of the pie” is constant and the division of that pie is of concern. Asmentioned, the nation that can influence terms of trade to obtain a bigger division will. This is of no concern,however, from a world point of view. What is of concern is when a nation does better under the two-stage gamethan under the one-stage game when the total stocks harvested are larger under that one-stage game. Thisscenario is possible - that one nation does better under the two-stage game but not both. In this case, the piecan be larger, so an international policy is required to obtain a Pareto-improvement. The policy consists of anagreement to partake in the one-stage game with a price mutually set that ensures that each nation does betterthan under the two-stage game. Under these circumstances, one nation effectively threatens the other intoimproved terms of trade for itself, and the world is better off regardless.132An alternative model is proposed where this production and trade game takes place over twostages rather than one. The second stage allows the possibility of influencing the trade terms andtrade price. Nations try to influence the trade terms through their first-stage productiondecisions. When the open access resource is unrestricted, the nations’ actions to influence thetrade terms results in an outcome that is less efficient than the Ricardian outcome. When theresource is restricted to a level below the combined autarkic harvesting capacity, no trade takesplace. When the resource level falls between these two points, trade occurs so that no resourceis wasted. In this region, a nation can affect the form of the production and trade structure inorder to gain a larger division of the excess resource stocks (than it would have in the one-stagegame).The analysis of these games has produced some interesting policy recommendations. Amongthem are: 1) R&D subsidies should be directed at industries where a nation has comparativeadvantage in order to achieve both an individual and joint welfare increase; and 2) a governmentshould attempt to control production and trade so as to structure trade as a two-stage game(whether open access resource stocks are restricted or not) if this structure increases its nation’sown welfare, regardless of the possibility of decreased joint welfare. Since there is nointernational enforcement against the latter policy, trade associations such as GATT may wantto consider regulations against it (unless decreased “fish” production is actually welfareimproving due to positive temporal effects of increased present stocks).133Future work may consider the temporal effects of the existence of the internationally open accessresource more explicitly in the existing model framework. Other future work would relax someof the strong assumptions about alternative uses of factor endowments and intermediate goods,and the number of factors and nations involved. An analysis of representative firms within thenations is also possible in the future. Perhaps with the addition of these elements, a number ofthis model’s conclusions could be tested empirically.134CHAPTER FOUR: OVERALL CONCLUSIONSThe three essays use standard economic tools to model some specific processes in the areas ofentrepreneurship, joint venturing, and trade. The processes are then examined for inefficiencies.Once the inefficiencies are understood, policy recommendations are offered to increase efficiencywhere possible. In the first essay, a policy is offered that would preclude the solution of aPrisoners’ Dilemma among incumbents through collusion when such collusion is to the detrimentof consumers and new entrants. In the second essay, a method is presented to solve a Prisoners’Dilemma between partners in a joint venture when cooperative completion of that venture isbeneficial both for the firms involved and for society as well. In the third essay, policies areoffered that would eliminate inefficient arms-race-type investments in order to increase worldwelfare.The policies that are generated by the three essays may be worth considering because they arebased on different analyses than found in the existing literature. The first essay uses slightlyaltered classical economic assumptions, game theory, and an incumbent’s perspective in orderto explain the emergence of entrepreneurship. Such an approach is somewhat novel to the field.The second essay provides solutions to the one-shot Prisoners’ Dilemma that are markedlydifferent from any found in the existing literature. As well, the specific application to R&D jointventures is unique. The third essay models the production and trade process in two periodsinstead of the standard one period found in most Ricardian trade literature. The essay alsoanalyzes trade for a wide range of open access resource levels, unlike some trade literature. As135a result of the distinctive nature of the essays, their recommendations may provide new insightinto important problems.The essays, although somewhat unique in their fields, do share some common ground amongstthemselves. First, they all involve some element of dynamics - they all model actions that takeplace over more than one period of time. In fact, elements of all the essays transform thestandard static model of each problem into a more dynamic one. For example, the first essayadds a second period to the competitive environment and this allows the possibility of newopportunities to be available to entrepreneurs. Second, all three essays involve technology atsome level. Essay one uses a technological advance to spur changes in the industry structure.Essay two includes technological complementarities to create the joint venture Prisoners’Dilemma. Essay three assumes trade based on comparative advantage in production technologies.So, although the three essays cover disparate topics, they share some common modellingelements.Now, consider the conclusions that are drawn from each essay separately.The first essay, “Technological Force: the Emergence of Entrepreneurship”, shows that, withslight changes in assumptions, classical economic theory does have room for entrepreneurship.Simply by adding a dynamic element to a classically-based model of competition and byassuming foreseeable innovations occur as time progresses, entrepreneurs do emerge in asubstantial range of the model’s area of existence. Further, analysis reveals that the outcome136created by entrepreneurial emergence can be welfare improving.It would appear that entrepreneurs emerge in the much of the relevant parameter-space of thegames defined in this essay. However, policy action can affect entrepreneurial emergence in theCo-ordination (CG) and the Prisoners’ Dilemma (PD) games. In the case of the CG, some smallsubsidy can be given to one or both incumbents to ensure that both do not remain flexible andblock entrepreneurial emergence, if that emergence is of net benefit to society. In the case ofthe PD, policies that discourage collusion (ensure competition) between the incumbents (andbetween all later firms) will help ensure the emergence of welfare-improving entrepreneurs.Many current competition policies only discourage collusion that has a material effect onindustry competitiveness from a price perspective, while allowing some industry cooperationwith respect to technology-sharing. This essay argues that competition policy should extend tothe technology strategy regime as it does to the price (or quantity) fixing regime becausecollusion on technology decisions could be socially damaging.This essay also answers an important question: Why, even when it might appear in the realworld that incumbents could do “better” by focusing on future opportunities and blocking theentry of a lot of the entrepreneurs that will end up destroying them, some incumbents choosenot to. The essay provides an alternative explanation to the incentive-difference-based patent-raceand technology-adoption literature and to the organizational inertia and boundedly-rationalincumbent literature. The essay’s explanation is based on a model of rational strategic choice toexplain the emergence of entrepreneurship.137The second essay, “An Auction Solution to the Joint Venture Prisoners’ Dilemma”, presentsimplementable solutions to a JVPD. Even when the payoffs are not completely transferable, anappropriately constructed Auction Solution may result in a Pareto-improvement to the JointVenture under certain parameter ordering restrictions and the availability of certain futuresmarkets. Therefore, policies that enable the Solutions to be implemented are encouraged whenany of these Solutions is the best way to solve the dilemma.The Auction Solution presented has many advantages. The auction mechanism is simple tounderstand, legal, and requires few resources (just a machine to hold bids, compare them, andthen distribute the shares and bids). It allows the optimal bidder (the one with the highestvaluation of the venture) to obtain ownership. The Auction Solution may also be more acceptableunder competition law. It does not have the same “overly-cooperative” appearance as contract-based scenarios (like the side-payment solution); after all, the auction is a competitive one. TheAuction Solution also appears to have the most flexibility for obtaining the cooperative outcometo any joint venture that can be represented by a Prisoners’ Dilemma, especially if it entailsstrategic non-transferable costs.The third essay, “Fish and Ships: Trade with Imperfect Competition and an International OpenAccess Resource”, analyzes trade when nations can exchange intermediate goods that allow themaccess to a final open access good. In a one-stage game when the open access resource stocksare not restricted, the nations are price-takers and a Ricardian world results. When that resourceis restricted to a level below that of combined autarkic harvesting capacity, no trade will take138place. When the resource level falls between these two points, trade occurs so that no resourcesare wasted. When trade occurs, both nations benefit (i.e., increase welfare over autarky).An alternative model is proposed where this production and trade game takes place over twostages rather than one. The second stage allows the possibility of influencing the trade terms andtrade price. Nations try to influence the trade terms through their first-stage productiondecisions. When the open access resource is unresthcted, the nations’ actions to influence thetrade terms in their favour results in an outcome that is less efficient than the Ricardian outcome.When the resource is restricted to a level below the combined autarkic harvesting capacity, notrade takes place. When the resource level falls between these two points, trade occurs so thatno resource is wasted. In this region, a nation can affect the form of the production and tradestructure in order to gain a larger division of the excess resource stocks.The analysis of these games has produced some interesting policy recommendations. Amongthem are: 1) R&D subsidies should be directed at industries where a nation has comparativeadvantage in order to achieve both an individual and joint welfare increase; and 2) a governmentshould attempt to control production and trade so as to structure trade as a two-stage game(whether open access resource stocks are restricted or not) if this structure increases its nation’sown welfare, regardless of the possibility of decreased joint welfare. Since there is nointernational enforcement against the latter policy, trade associations such as GATT may wantto consider regulations against it (unless decreased “fish” production is actually welfareimproving due to positive temporal effects of increased present stocks).139These three essays offer some new and potentially valuable insights in their specific areas ofeconomics. When analyzing a problem in any area, the ability to view (and model) an issue fromdifferent perspectives allows improved solutions to be generated (and implemented). 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As well, d > 0 is assumed to allow incumbents to be profitable underthe dominant strategy outcome. The ordering and the positive profit condition define aPrisoners’ Dilemma outcome where lock-in, L, is the dominant strategy played by theincumbents. As a result, new entrants do emerge in period two.The conditions translate to the following:(A c)2d>0= ‘ -f>O9B(A+c —2c1)2 (A—c)2 (A—c)2w> c-f9B 9B B(n÷l)2147(A— c )2 (A— c )2 (A — c,)2c>d= f-f+ -f> -f9B B(n÷1)2 9B(A—c1)2 (A÷c1—2c)2 (A—c)2d>s=-f> -f+ ?‘ -f9B 9B B(n+1)2Note: it is implicitly assumed above that no incumbent who chooses to lock-in in period onecan profitably remain in the market in period two (see Proposition One of Chapter One).2. For the Mixed Strategy outcome of the Co-ordination Game, the ordering w > c ands > d is required. As well, d > 0 is assumed to hold as before. See above formathematics of restrictions.3. The “Co-ordination Game” where players want to play L if other plays L and F if theother plays F is ruled out because if c > w then it is impossible for d > s (seeAppendix Five for details).4. A Pure Strategy Equilibrium outcome occurs when the dominant strategy corresponds tothe individually efficient firm strategy. There are two possible:1) when c > d and c > w then F is the dominant and individually efficient firmstrategy. No new entrant with new technology (a technology not also implementedby an incumbent) emerges;1482) and when d> c and d> s then L is the dominant and individually efficient firmstrategy. It is ensured by Proposition One of Chapter One that entrepreneursemerge in period two in this case.149Appendix Two: Numerical Examples of the Normal Form Game0.78 0.250.78 1.001.00 0.360.25 0.361.21 0.661.21 1.271.27 0.610.66 0.61Figure Seven: Numerical Examples of the Normal Form GameNumerical Examples of the Normal Form Game:Common assumptions for all following games:Fixed costs (f = 3), Variable costs for flexible incumbent (cf = 4),Variable costs for locked-in incumbent (ci = 3.5)Variable costs for locked-in new entrant (cn = i5Inverse demand function (P = A - B*Q, , where B = 1)Prisoners’ Dilemma Mixed Strategy Outcome Pure Strategy OutcomeOutcome12flexible lock-inA=9.1 A=9.3 A=1012flexible lock-inf1eIx110Ck12flexible lock-inI1f1ex10Ckf1eIx110Ck1.04 0.411.04 2.292.29 1.550.41 1.55Note: read payoffs in cells as: Player One payoffPlayer Two payoff150Annendix Three: First Period Entry RestrictionsRestrictions that ensure both incumbents are profitable when they both choose eitherflexible, F, or lock-in, L, technology strategies in period one are:i) In the case of both incumbents locking-in in period one:(A c)2LL,LL>O9 B’f> 0ii) In the case of both incumbents being flexible in period one:(A —c)2 (A —c)2F>0 I ‘-f>0FF,FF 9B B(n÷1)2where:(A - c)n = IN1EGER[ “ - 1] 2where : IN7EGER + operator returns the largestinteger composition of its contents zfpositive, otherwise it returns zero.2. Restrictions that ensure no “third incumbent” in the first period are:i) For the all incumbents locked-in case:(A-c)2ItLLL,LLL <0 16 B -<0ii) For the case where third incumbent is flexible, and the others lock-in:151A+2c —3c)2 A—c)2LLF,LLL <0 16’ B - + B (n + 1)2 - f< oiii) For the case where the third incumbent and one other is flexible, while theremaining incumbent locks-in:(A + —2c)2 (A —c)2LFF,LLL < 16 B - + B (ii + 1)2-<0iv) For the case where all incumbents are flexible in period one:(A—c)2 (A—c)2FFF,LU<° 16B B(fl÷1)2v) For the case where third incumbent locks-in and the two other incumbentschoose to be flexible:(A + 2 c — 3 c)2It L<O I -f<0FFL,LLL 16 Bvi) For the case where the third incumbent and one other incumbent locks-in whilethe remaining incumbent chooses to be flexible:(A + c — 2 c1)2L<0 f-f<0FLL,LLL 16 Bwhich is subsumed in v) above.Note: it is implicitly assumed in 1. and 2. above that no incumbent who chooses to lock-in inperiod one can profitably remain in the market in period two (see Proposition One ofChapter One).152Appendix Four: Division of Parameter Space by Game-TypeFigure Eight: Effect of Changes in Elasticity on Parameter Space DivisionNote: Game-Type Legend:PD = Prisoners’ Dilemma normal formCG = Co-ordination game normal formFD = F-dominant Strategy normal formLD = L-dominant Strategy normal formParameter Space Game Divisioninto PD, CG, FD or LD games100f3, Cn1; Cf, CI, A varying.90a)a 80a4-c 70cl)C.)Ca)a>C.)Ca)U201001 2 3 4Type of Game 1 PD, 2CG, 3FD, 4=LDB=0.5 8=3.0 8=5.5153B1, Cn1; Cf, Cl, A varying.Figure Nine: Effect of Changes in Fixed Costs on Parameter Space DivisionParameterinto PD,1 00Space Game DivisionCG, FD or LD games90Q)o 800-Ic 701)1)0>..C-)C11)020U1001 2 3 4Type of Game 1PD, 2CG, 3zFD, 4=LDf=0.5 f=3.0 fjf=5.5154ADDendix Five: Accounting of Possible GamesThere are four different payoffs possible in a two by two normal form game with two symmetricplayers, and they have been labelled w, c, d, s. Consider the non-pathological cases wheninequalities between pairs of these payoffs occur (note: pathological cases occur when two ormore of these payoffs are equal). There are 4-choose-2 cases to consider:c>s, w>d, c>w, c>d, w>s, w>c, s>d, s>w, d>c, d>s, s>c, d>wThe last two of these inequalities cannot hold given the actual construction of the payoffs.—=(A—cf)2—(A +c1—2 p29BNote that the contents of the two round brackets must both be positive. Examine which of thesecontents is larger:(A — c) — (A ÷ c1 —2 c? = C1 — c,> 0 as C1> C1Therefore c > s by construction of the payoffs.A similar examination can be done for d> w:(A + c — 2 c1)2 — (A —c1)2w—d=9B155This simplifies to:(c,— c1) (2A —3 c1 + c)f >OasA>c1c,9BTherefore w> d by construction of the payoffs.Consider now all the possible ways of ordering the four payoffs. There are 4-factorial ways todo so:Orderingw>c>d>sw>c>s>dw>d>c>sw>d>s>cw>s>c>dw>s>d>cc>w>d>sc>w>s>dc>d>w>sc>d>s>wc>s>w>dc>s>d>wd>c>w>sd>c>s>wGame TypePrisoners’ DilemmaCo-ordination GameL Dominantruled out as c> s byruled out as c>s byruled out as c>s byCo-ordination GameF Dominantruled out asruled out asF Dominantruled out asruled out asruled out asw>dbyw>d byw>dby156Note12constructionconstructionconstruction(ruled out) 3constructionconstructionconstructionconstructionconstructionw>d byw>d byNotes:Formally, a Prisoners’ Dilemma game may also include the restriction on payoffs that2c > w + s to ensure that the Pareto-optimal outcome for the players is to play F ineach game and is not to take turns playing L on the other’s playing F. Players areeliminated in the next period after they play L, therefore, they are not available in thenext period to play F and return the exploitative favour. As well, only agreements thatwould not adversely affect market forces could be used to enforce such mutualexploitation. The ordering of w> c> d> s reveals that incumbents could increase theirindividual (and joint) welfare if they could agree to collude. The further restriction that2c > w + s does not affect the other entry and welfare effects presented in this paper.d>w>s>c ruled out as w>d by constructiond> w > c > s ruled out as w> d by constructiond>s>c>w ruled out as w>d by constructiond>s>w>c ruled out as w>d by constructions>w>c>d ruled out as c>s by constructions>w>d>c ruled out as c>s by constructions> c> w> d ruled out as c > s by constructions>c>d>w ruled out as c>s by constructions>d>c>w ruled out as c>s by constructions>d>w>c ruled out as c>s by construction1572. This Co-ordination game lies on the asymmetric diagonal. When its rival incumbent playsF, the incumbent wants to play L; when that rival incumbent plays L, the incumbentwants to play F. As w > s in this game, both incumbents want to play L to the other’splay of F. Hence, there is a co-ordination problem between the incumbents. This maybe overcome through some form of collusion. This essay suggests that a policy thatwould help coordinate the outcome would be valuable. A policy that provides either (orpossibly both’) incumbents an incentive to play L would provide an increase in is known that there is an increase in welfare when the (F,F) outcome is displaced by either the (L,L) outcomeor the (F,L) (or (L,F)) outcomes:LL-FF =(_I) [(A - c1)2 - (A- c1)2] > 0AicFLFF = (cf - c1) (2A + 3 c1 -5 c1) (-i-) >0IXCSLLFF = (—--) [(A — c,)2 — (A — c)2] > 0c 3c 1IXCSFLFF = (c1 - c1) (2A --- -j-t) (-) >01583. This Co-ordination game is ruled out by construction of a set of payoff relationships.When c> w it is impossible for d> s:(A— c)2 (c1 — c1) (4 A — 4 c,)c>wB(n÷l)2 9B(cf—cj)(4A—4c) (A—c)2d>s=-f9B B(n+1)2In order to satisfy these two conditions then:4A-4c,>4A-4c1—c>BUTKNOWTherefore, this Co-ordination game is ruled out by construction of the payoffs.159Arnendix Six: Existence of Prisoners’ DilemmaIt can be proven that the possibility a Prisoners’ Dilemma normal form will never occur underall parameter values is zero.Given that:(1) A > Cf > c1 > c > 0: the parameter ordering in this model(2) n 2 in period one : non-negative profits for the two incumbents given the lowestsymmetrical payoff [here, assuming the PD where c> d,non-negative profits must be available to incumbents underthe (d,d) scenario]The Prisoners’ Dilemma normal form then requires:(3) d> 0 : non-negative profits for incumbents under the Nash equilibrium outcome(4) w>c(5) c>d(6) d>s(A—c,) (A—c1)(3) >f while (2) —______- 1 29BThe fixed cost factor can be defined in terms of other parameters to satisfy both conditions:(A c)2setf=16B(3):4>3,(2):32Now compare (4) to (6). It can be seen that both place a restriction of the same form on thepayoffs. However, the restriction of (4) is subsumed in the restriction of (6). Therefore, with160f set, only two more inequalities need to be satisfied: (5) and (6). Consider (5) and (6).Satisfying both inequalities reduces to satisfying one of them the condition that:2A >3 c - c,which is> Oas c1> c1The inverse demand function intercept, A, can then be defined in terms of another factor in thegame as: A = 4 Cf. This satisfies the condition above.Now consider satisfying (6). Substituting in the definitions of A and f, this reduces to:4 (4 c — c) (16)(9)(4 c — c )2{ IN7EGER [ ‘ ‘z f(4 Cf — c,) (64)(c, — c,)(3 Cf) + (9)(4 Cf — c,)2Define c1 in terms of another factor in the game as: c1 = Cf!2. Substitute this definition into thepreceding equation. Now, in order to satisfy all the conditions to obtain a Prisoners’ Dilemmanormal form as described above, one final equation needs to be satisfied through one parameter-ratio:C CIN7EGER[4.57 - 1.14—J > 2.70 - 0.68 —Cf CfGiven that Cf> C, the range of the ratio c/cf can be from 0 to 1. The LHS of the equation thenhas the range 4 to 3 with a corresponding range to the RHS of 2.7 to 2.0. It can be seen thatthe equation is satisfied under all of the range without having to define the parameters further.Therefore, given any Cf value, a large range of parameter values can be found that will satisfythe Prisoners’ Dilemma normal form. Thus, the Prisoners’ Dilemma normal form subset of thefull parameter set is non-empty.161ADDendix Seven: Table of SolutionsSolution Type Advantages DisadvantagesEx-Ante Auction Pareto-optimal outcome is possible. Requires payoffs be transferable to work; orPossible to use when some payoffs are non- some restrictions on transferable versus nontransferable. transferable payoffs (so may require specialNo third-party strategy verification is required. futures contracts).Ownership goes to the optimal firm when firmsare asymmetric.Coin-Flip Pareto-optimal outcome is possible. Requires payoffs be transferable to work; orRelatively simple to implement. payoffs are made transferable through specialNo third-party strategy verification required. futures contracts.Results in expected value payoffs only.There is a loser who ends up with nothing.Full Contract Payoffs need not be transferable. Minimum of two third-party strategyPareto-optimal outcome is possible. verifications required.Relatively simple to implement. Need a legal system.Side Payment Payoffs need not be transferable. Minimum of one third-party strategy verification(no payoff transfers) Pareto-improvement is possible. required.Relatively simple to implement. Need a legal system.Entails restrictions on payoffs.Side Payment Pareto-optimal outcome is possible. Minimum of one third-party strategy verification(payoffs transferable) Relatively simple to implement. required. Need a legal system.Entails transferability and restrictions onpayoffs.Each player has no incentive to initiate thesolution.Transference Pareto-optimal outcome is possible. It is a merger.Relatively simple to implement, Requires special futures contracts.No third-party strategy verification required. The joint venture product is split up.Not possible to use when any payoffs are nontransferable.same rour: same oi oiuuous to use.j vri162Appendix Eight: The Coin Flip SolutionAlthough the Coin Flip Solution may not choose the optimal winner in the case of asymmetricfirms, it is nonetheless an interesting alternative solution to consider. It is a solution which isvery simple to implement but has some penalties if firms are at all risk-averse.In order to analyze this Solution consider the basic application to two symmetric firms asdescribed in the Auction Solution base application. Here, the two firms agree to partake in thesolution, then they put their shares in the trust and have the trust flip the coin without revealingthe winner at that time. The firms then play their venture strategies. The outcome of the ventureis then realised. The outcome of the coin flip is revealed with all shares going to the winner andnothing going to the loser. With a fair coin it can be seen that each firm foresees an expectedvalue of one-half the full value of the venture outcome. Each firm must consider that this rewardis dependent on its venture strategy as well as on that of the other firm. Thus, each firm can seepursuit of the same goal - to maximize the venture outcome- as being in its own self-interest.Consider the strategies that firms would play to maximize the value of the venture outcome (seeTable Five for details2). If one firm believed that the other would cooperate then it would alsocooperate in order to maximize the venture outcome because c > (w+s)12. Thus, mutualcooperation is a Nash Equilibrium in this scenario.2 Net payoffs referred to in this Appendix are not net the Solution set-up cost, 7/2, unless specified.163Now consider what occurs when one firm believes that the other will defect. If the expectedvalue of (w+s)12 is greater than the expected value of d then the firm should cooperate tomaximize the venture outcome. If payoffs are ordered as such (although this is not part of thedefmition of a PD) then mutual cooperation is the only Nash Equilibrium. If, however, thepayoffs are (w+s)12 < d then a forward induction argument can be used to eliminate the secondNash Equilibrium of mutual defection given that a -y--costly Coin Flip solution was entered.Consider risk-neutral firms. If the firms do not enter into the Coin Flip Solution they guaranteethemselves d as their actual net payoff whereas if they enter and defect then they receive eitherd-y/2 or (w+s-y)12 as their payoff, which are both less than d by assumption or defmition. Ifthe firms are at all risk-averse then a similar argument occurs because then the certain payoffby not entering the solution dominates the expected payoff (of d) of entering the Coin Flipsolution and defecting.Thus, the Coin Flip Solution can solve the JVPD in certain domains. Additionally, it involvesless complication than the Auction Solution. But it has one major drawback besides beinginefficient in the asymmetric firm case. It deals strictly with expected values. When firms areat all risk-averse (with respect to the payoff values3)then these expected rewards are less thanthe certain rewards offered by the Auction Solution. If, in the extreme case, the firms are so riskaverse to make the expected value of c worth less than the certain value of d then the Coin FlipSolution may not be attractive at all (while the Auction Solution still would be).Risk aversion here is defined in terms of the payoff values. Under the Coin Flip Solution, the firms risk thepossibility of ending up with a zero or negative payoff (if they lose the coin flip). In comparison, under theAuction Solution each player ends up with the same payoff (whether in terms of money or ownership of theventure) with certainty.164Item \ Scenario Mutual Cooperation Mutual Defection Single DefectionNet Payoff if 2c 2d w+swin the tossNet Payoff if 0 0 0lose the tossExpected Net c d (w+s)12PayoffTable Five: Payoffs under the Coin Flip SolutionThere is one other solution available that is as seemingly simple as the Coin-Flip Solution butwithout the problems associated with the all-or-nothing outcome, and that is to have a trustdivide all proceeds of the JV as they come in. For this to work, however, a number ofconditions must be satisfied. A PD must exist without SNTCs that is different from the onedescribed in the basic application in essay two. Unfortunately, this would entail the JV itselfgenerating revenues4.In the context of a R&D JV it is possible to generate licensing fees, forexample, but then it is improbable that a PD exists because one JV partner would have to havethe ability to take more than its fair share of those revenues under the original ownershipstructure. The case where such a trust could exist under the original application is tackled earlyin essay two, and the argument is made that any such institution which gives the partnersownership with certainty cannot work.In the usual case a JV would create an output (i.e., a new manufacturing process) to be used by the partnersin production of their goods. There is no method for one partner to take more than the other in this situation,so a PD does not exist and the trust is useless.165Anpendix Nine: Alternative Solutions Under Special Futures Contracts1. The Transference SolutionAssume the basic case described above under the SNTC section. Consider what occurs wheneach firm is given the special futures contracts of the other firm in exchange for its own.Specifically, consider the case where each firm is given contracts covering half the stock of theother firm at a transaction cost of ‘y!2. Each firm will now receive half of its individual netpayoff plus half of the other firm’s. The game is transformed into one where mutual cooperationis a Nash Equilibrium. If both firms cooperate, each receives c12+c12 = c. If both firms defect,each receives d/2 + d12 = d. If one firm defects while the other cooperates, the defector receivesw12+s/2 = (w+s)12 and the cooperator receives s/2+w/2 = (w+s)/2.If a firm believes that the other will cooperate, its best reply is cooperation as c > (w+s)12 bydefinition of the Prisoners’ Dilemma. Thus, mutual cooperation is now a Nash Equilibrium. Ifa firm believes that the other will defect, then its best reply will depend on the payoff fromdefecting versus cooperating. If (w+s)12 > d then cooperating is the best reply. In this case theonly Nash Equilibrium of the game is mutual cooperation and the Prisoners’ Dilemma is solved.If, however, d (w+s)12 then defecting is the best reply (or an indifferent reply in the caseof an equality). Mutual defection becomes a second Nash Equilibrium to the game. However,this Equilibrium can be eliminated by a forward induction argument. The payoff from mutualdefection (d > d-’y/2) can be achieved by not entering the Transference Solution and justcarrying out the original venture. By choosing to enter the solution, the only rational strategyto play is cooperation. The only Nash Equilibrium which remains after the forward induction166argument is made in either payoff ordering case is mutual cooperation. Thus, the TransferenceSolution solves the Prisoners’ Dilemma.This solution is just a simple merger limited to the venture itself. As it is known that a mergersolves the PD it is no surprise that this solution technique does so as well.2. The New Coin Flip SolutionNow, consider the next simplest form of the solution to the Joint Venture Prisoners’ Dilemma(JVPD) under SNTCs: a y-costly coin toss for ownership of the sum of each (of the two)participant’s net values of the completed venture. The unbreakable agreement to perform thecoin toss must be made before the venture is started, and the coin toss outcome revealed onlyafter the venture is completed. Otherwise investment strategies may be misplayed if theagreement is made after the venture is started, or strategies may be changed if the outcome ofthe coin toss is known before the venture is completed. For example, if the loser knew it wasthe loser before the venture was finished it would choose to invest its least costly resources intothe venture at that time.SNTCs are made transferable by issuing special futures contracts as described in theTransference Solution. However, in this case the special futures contracts cover the full stockof each participant, and are put into the prize pot (along with the shares of the ownership of theJoint Venture) to make up the prize.167When the New Coin Flip Solution is implemented, the original game of certain payoffs is nowtransformed into one of expected payoffs. As before, in any outcome each firm will receive anexpected payoff of half of the sum of the each participant’s net payoff of the completed jointventure (opportunity costs and spillover benefits accounted for). Thus, if both firms cooperate,each receives an expected payoff of c-’y12. If one firm defects while the other cooperates, eachfirm receives an expected payoff of (w+s-y)/2. If both firms defect, each firm receives anexpected payoff of d-y12.As in the case of the Transference Solution, the best reply to cooperation is cooperation and todefection depends on the relative values of d to (w+s)/2. Thus, if (w+s)/2 > d then the onlyNash Equilibrium is mutual cooperation. If d (w+s)/2 then mutual cooperation and mutualdefection are both Nash Equilibria. However, if the forward induction argument is againconsidered then the only Nash Equilibrium which results is that of mutual cooperation.Therefore, the New Coin Flip Solution results in an outcome that is jointly efficient.Although the New Coin Flip Solution is viable, it is not very realistic. Risk averse stakeholderswould probably not allow companies to partake in its implementation. Fortunately, there is analternative which generates similar or further efficiency-improvements (in certain regimes whenagents are risk averse). This alternative is the ex-ante auction, and it is somewhat morecomplicated to administer (see Section 3.5.1 of Chapter Two for details).168Aunendix Ten: Analysis of SNTC Solutions and Some Solution Extensions1. Comparison of SNTC SolutionsWhy consider the New Auction Solution when the Transference and the New Coin-Flip Solutionsare less complicated? The New Auction Solution can handle some cases of non-transferable costswhereas the other solutions cannot. The New Auction Solution has an advantage over the NewCoin-Flip Solution because it deals with certain payoffs instead of expected payoffs; this isvaluable when stakeholders are risk-averse. The New Auction Solution is more competitive thaneither a simple coin-flip or a simple exchange of special futures (because the exchange ofownership is based on a competition - an auction); this is of value when anti-trust considerationsare important. The New Auction Solution is better than the Transference Solution when theoutput of the Joint Venture itself is difficult to split up or has less value when split up. The NewAuction Solution does not split up the output whereas the Transference Solution does. There isone other important reason why the New Auction Solution may be a superior one but lies in anextension of the model to be explored next.2. Extensions to the SNTC SolutionsNow consider some extensions to the model of SNTC. First, consider what occurs when thenumber of players increases from two upwards. Second, consider what occurs when some nontransferable costs enter the model.1692.1 Extension to More Than Two FirmsJust consider the three Solutions - Transference, New Coin-Flip and New Auction - in thisanalysis. To accommodate n (where n> 2) participants in the Joint Venture under Transferencesimply allow each player to exchange 1/n of its own special futures for 1/n of each of theother’s. After the exchange each firm owns 1/n of each of the n firm’s special futures (includingits own). Then each firm becomes, in effect, an equal partner in the net payoff to the allparticipants in venture. Mutual cooperation may then be achieved as described before.Under the New Coin-Flip Solution n players are accommodated by using a fair n-outcomegenerator instead of one simple coin-toss. Again mutual cooperation may then be achieved asdescribed before.Under the New Auction Solution the n participants must follow the same procedures as the twoparticipants did before. The only change necessarily required is to use some fair n-outcomegenerator in case of a tie in the bids (which is the equilibrium case). The other change that maybe desired is to alter a to allow bids to reflect the total value. To do this, it is necessary to havea = 1/112. Mutual cooperation may then be achieved as described before.This extension gives one more reason to use the New Auction Solution over the TransferenceSolution and the New Coin Flip Solution. If a firm’s own special futures have some controlpremium5 then the New Auction Solution is most efficient in re-issuing those contracts. TheControl premium here means that there exists some value over the face value to controlling the futures contractsto the firm issuing that stock itself. For example, these contracts may entail some persuasive valuable powerinside the company, or some voting influence, or be of some importance to the board of the company as a signal170auction winner can give each firm back its special futures as compensation (for example,consider that this is required under the rules of partaking in the Auction Solution). UnderTransference firms can exchange their special futures to gain their own back. These ex-postexchanges differ in their number depending on whether the New Auction or the TransferenceSolution is used. Under the auction, only n - 1 deals need to be done (the winner deals witheveryone else). Under Transference, [n (n- 1)1/2 deals need to be done (which is a greaternumber of deals when n>2). Thus, if there is some cost to these ex-post transactions then theNew Auction Solution is superior, all else being equal. Under the New Coin Flip Solution, ofcourse, no contracts are returned to the losers, so their control premium is lost.2.2 The Non-Transferable Costs ExtensionThe Transference, New Coin-Flip and New Auction Solutions are based on the JVPD payoffsbeing transferable through the special futures contracts. This requirement leads to an interestingresult regarding the symmetry of the participating firms. When the JVPD payoffs are consideredcompletely transferable the result is that the players have symmetry forced upon them. Theplayers have the same payoffs, the same strategy space, and the same rationality. Therefore, theyare symmetric.Relaxing this symmetry requirement provides a practical extension to the models. Under thefully transferable case, if the net payoffs from participating in the joint venture initially differamong firms under similar outcomes then the three Solutions provide the same net payoffs toto the market.171each firm in equilibrium regardless. Such a result may not seem fair to the firm who initiallywas gaining more than its rivals under any particular outcome. Although the treatment ofasymmetrical players is left for future work, it appears to be possible to achieve fairer outcomesfrom the Solutions by altering the amount of special futures given by each participant so as toreflect the asymmetry. For example, a firm that was gaining more from the venture output thanits partners could, through negotiating a “fair” agreement (i.e., where all parties receive payoffsonly according to their own potential), only have to give up a fraction of special futurescontracts that the other firms do. If firm A generated twice the value from the venture’s outputas B, a fair requirement may be to have A only give B half the special futures contracts that Bgives A so each exchanges goods of the same total value. Leaving formal analysis of thissymmetry consideration for future work, the extension to non-transferable payoffs is nowexplored.When the JVPD net payoffs are not completely transferable some of the three Solutions may notapply. First, consider symmetric players with the same non-transferable payoffs6.These payoffswill weigh on a firm’s evaluation of its net payoffs under each scenario in the dilemma. It isassumed that the non-transferable payoffs are worse for cooperation than for defection.Section 3.4 of Chapter Two provides the basis for how this case is handled. The non-transferablepayoffs are simply SNTCs of that section. Only under certain parameter restrictions will a nearPareto-optimal solution be available. Oniy a redefined New Auction Solution will be applicable;6 For example, if the principals involved in the participating firms achieve some personal level of satisfactionfrom cheating and some personal dissatisfaction when cheated on then these cannot be transferred even by thespecial futures contracts.172the other two solutions will not work in this case.A further, more complicated extension than the one just explored could include non-symmetricopportunity costs (or benefits). It is hypothesized that there would be a significant range of thesegames whose outcome could be improved upon by the implementation of an appropriatelydefined Solution.173Appendix Eleven: Equilibria with Restricted BiddingFirms caii only bid either 2c, w+s, or 2d under restricted bidding. They can play mixed bidsof this choice set. The following analysis reveals how each firm will choose its best mix andstrategy.Consider Firm A’s best bid mix when it assumes Firm B will play C and bid 2c with xprobability and w+s with (1 - x) probability. First, consider A cooperating and bidding 2c withprobability y and w+s with probability (1- y). The calculation simplifies to:Max c w+s)+—j y=1The optimal choice is to bid 2c with certainty, bidding consistent with strategy played and beliefof the other firm’s strategy played.Similarly, when A considers defecting with the same beliefs on B as above, A’s optimal bid willbe consistent with its play: A will bid w+s.The next step is then to compare the payoffs generated by each strategy and choose the dominantone. Assuming B plays C with certainty, and bids 2c with probability x and w+s withprobability (1- x) gives Firm A the following payoff comparison between cooperating anddefecting (assuming it bids optimally):174C W+S . C W+S W+S(1—x)(— )+c—z( VS• (_ )x+ 2The required restriction on i (see Section 3.4 of Chapter Two) can then be taken from thiscomparison to ensure that mutual cooperation is a Nash Equilibrium under the restricted bidsauction.Similar analysis can also be done when it is assumed that B defects to find the condition on L(see Section 3.4 of Chapter Two) that makes mutual defection a Nash Equilibrium under therestricted bids auction.175Annendix Twelve: The Nash Bargaining SolutionThere is another literature which is relevant to the models presented in this paper - thebargaining literature. Although the Walrasian market clearing solution is assumed to be themechanism that determines the terms of trade (the price) in both the one and two-stage game,that solution corresponds to a bargaining solution. Whereas a Walrasian auctioneer may notappeal to some as a realistic mechanism, bargaining may be more widely accepted (and isanalyzed here). Furthermore, the bargaining mechanism provides an alternative exposition ofhow different parameters and strategies can affect terms of trade, for example, through movingthe threat points (i.e., bargaining positions) of the nations involved.The relevant bargaining literature begins with Nash’s (1950 and 1953) work on the bargainingsolution. Rochet (1987) provides a comprehensive review of initial and recent developments inthe field since Nash. Osborne and Rubinstein’s (1990) text presents a good analysis of basicbargaining and extensions. Recent papers focusing specifically on international exchange includeChan (1988) and Rogoff (1990). Rogoff finds: 1) that when nations bargain over utility(including risk aversion effects) the nation with the higher threat point gains more from thetrade; and 2) that nations will change investment patterns in order to enhance their bargainingpositions. Shogren (1992) finds evidence against Rogoff’s first claim. Shogren’s experimentsreveal that negotiated agreements usually split rewards evenly (85% of the time). This tradepaper assumes a Walrasian market clearing solution which corresponds to the Nash Bargainingtype Solution outlined below.176Assume that the nations bargain over the division of the two intermediate goods. The productof the two intermediate goods stocks of a nation is used as the proxy for harvesting capacity forthat nation. The negotiation process results in an equal gain in this harvesting-capacity-proxy.The gain is measured from a nation’s threat point (or harvesting-capacity-proxy evaluated at theend of stage one - when the intermediate goods have been produced).The proxy is used rather than the actual harvesting capacity because the Nash BargainingSolution only applies (in general) when the function representing the welfare-generating-goodsbeing bargained over meets an invariance criterion (i.e., if the amount of welfare-generating-goods is linearly transformed then the new bargaining solution is the linear transformation of theold bargaining solution). The harvesting-capacity-proxy (i.e., the product of good one and goodtwo) is such an invariant function. The Cobb-Douglas function (or the log of that Cobb-Douglasfunction) does not represent such an invariant function as non-linear transformations are involved(i.e., roots or logs).Consider the following representation of the Nash Bargaining Solution to the negotiations overtrading of harvesting-capacity-proxy elements:Max1, : [(gm — t1) (g2A + )—g g] [(glB + t1) (s — )— glB g2Blwhere: g = Nation J’s product of good i just before trade= the amount of good i traded among nationsIf the only factor of welfare-generating-goods, H, is linearly transformed then the bargainingsolution (t1, t2) defined by the maximization above is as well:177Consider H’ = z H + (3this can be written as H’=y H instead without loss of generalityihe solution of the Maximization above is:— 18M + gIA — g2A g12— “g + g2B — 2 (g ÷ giB)as g, = H multiplied by some productivity-type constants it canbe seen by inspection that if the transformation is done thenH’=y H =g’ = yg t/= y t1so the invariance criterion is met.However, by inspection, it can be seen that if the actual harvesting capacity function were usedinstead of the proxy then the invariance criterion would not be met. Thus, the proxy is used indetermining the final terms of trade bargained to. The threat points are clearly defmed as thepre-trade product of the two intermediate goods each nation has.The harvesting-capacity-proxy assumed here needs to be adjusted when the importance of goodone and good two differ (i.e., when a 1/2). A weight adjustment is factored into the amountof good two exchanged for good one when a 1/2 . The Nash Bargaining Solution (NBS)equation for the price negotiated, p, becomes:178psuchthat:W -w =w -wAafter trade Apre_trade but after stage one 8qfter trade Bp,e_gj but after stage onewhere: W is the harvesting-capacity-proxy (proxy for Y)after tr- = (TM (H — xA) — t) (r14 XA +(1— a)W =(r (H—x))(r x) but after stage one IA A 14 AWRafter = (r (H — xB) + t) (r,3 x2 — p (1 a) t)W =(r (H-x))(r xB)Bpre.tra& but after stage one lB B 2Bwith [‘_-_“1 as the weighting adjustment when aa 2It is interesting to note that the resulting equilibrium from the negotiation process outlined abovefalls between two other Nash Bargaining-type Solutions: The first of which consists of thenations sharing equally in harvesting capacity gains (not proxy gains) from trade over theautarkic threat points. The second of which is based on same weight-adjusted intermediate goodharvesting-capacity-proxies as outlined above. However, the nations share equally in harvesting-capacity-proxy gains measured from their autarldc threat points (i.e., not their threat points justbefore trade, that is, after stage one in the two stage game). On and between the two NashBargaining-type Solutions is where experimental bargaining outcomes have been found to lie.Thus, the equilibrium concept assumed above corresponds well with experimental evidence suchas that presented in Roth, Malouf and Murnighan (1981).179Anpendix Thirteen: Analysis of Case iii. Under Stackelbera LeadershipConsider the two-stage game if the first stage had a Stackelberg leader. The leader increases itsown harvesting capacity by increasing production where it has a comparative advantage to alevel over that dictated by the optimal (simultaneous) solution7. For example, if Nation Aunilaterally committed to an XA > XA and Nation B produced a best response to this, NationA would increase its harvesting capacity (as would Nation B).When a nation does increase specialization as a Stackelberg leader, the follower Nation does soas well in a best response. The follower, however, obtains a larger increase in harvestingcapacity than the leader. The follower profits more because it has the final opportunity to alterprice and trade level through its own production decision based on that of the leader. Thefollower specializes less than if it were the leader, and so influences price to its advantage. Bothnations gain when there is a Stackelberg leader, but the equilibrium joint harvesting capacity stillmay fall below that of the one-stage game regardless of which nation is the leader, as thefollowing numerical example shows (based on the numerical example thus far assumed in thethird essay):This phenomenon only occurs for increases up to the efficient division of production for the world (e.g., NationB has XB = H).180When Nation A leads: A = 2.074 = 2.006iXY = 0.118 AY = 0.194AversIAs autarky Bversus autarkyWhen Nation B leads: A = 2.113 = 1.950iXY = 0.157 tY = 0.138Aversus autarky Bversus autarkyNow consider the situation depicted in Figure Ten. The best response curve of nation 1 in thediagram, Ri, shows the specialization decision that maximizes l’s welfare for everyspecialization decision of nation 2 (and similarly for R2). The constant utility curve for eachnation (shown as a broken line) pealcs on that nation’s best response line (i.e., nation l’sconstant utility curve is horizontal when it touches nation l’s best response curve). With respectto the two-stage game in this paper, more specialization translates to nation A decreasing XAtowards 0 and nation B increasing xB towards H, and utility translates to harvesting capacity.When there is no Stackelberg leader, the two-stage game equilibrium is where the two bestresponse curves meet, at N. When nation 1 is the leader, the equilibrium shifts out to the pointSi where the constant utility curve of 1 is tangent to 2’s best response curve. At Si nation 1 ismore specialized than at N, as is nation 2. However, because of the (less than 45 degree) slopeof 2’s best response curve, nation 2 does not increase specialization as much as 1. The converseis true at S2 when nation 2 is the Stackelberg leader.Now consider the terms of trade as determined by the Nash Bargaining Solution to theharvesting-capacity-proxy as outlined in Appendix Twelve. When a nation increases itsspecialization, it decreases is proxy threat-point (or position to leverage the terms of trade in the181Figure Ten: Stackelberg Leadership in the Two-Stage GameWairasian case) before trading. Thus, when a nation is the Stackelberg leader, and so increasesits specialization more than the follower, it can expect to reap less of the gains from trade thanthe follower as it has a weaker threat point (i.e., lower price elasticity of trade). Of course, bothnations do benefit from the leadership as both specialize more, given specialization is a strategiccomplement.S2The Effects of Stackelberg Leadership on EquilibriumIn the Two-Stage Game:K2 RiR2a)U)4)curve for 2KiIncreasing Specialization for 1 I182ADnendix Fourteen: Proof of Specialization in One-Stage Game with Unrestricted StocksHypothesis: At least one nation fully specializes its production when nations trade in a one-stage game under the assumption of unrestricted open access (final) good stocks.Proof: In order for Nation A to fully specialize given any production decision of NationB requires that the following first order condition holds true (with the secondorder condition satisfied for a maximum):8YA r H (1 — a) (rlB + TM)—IaXA. XarM r28 a + T1B T2A (1 — a)Similarly, in order for Nation B to fully specialize given any production decisionof Nation A requires that the following first order condition hold true (with thesecond order condition satisfied for a maximum):T2B H (rM (1 — a) — T1B a)—I O=xaXB- H rM r2B (1 — a) + rlB r14 aIf XA as defined by (3-5) is greater than or equal to zero, then all that is neededto have B fully specialize is for the optimal production division response of A tothe conditions assumed to be less than or equal to the production division requiredby the first order condition outlined above, or that: XA° - 0. Recall thatmore specialization for A means a smaller production division, XA. Therefore, the183condition where the optimal response of A allows full specialization by B is:a H (1 — a) (TM T2B — r r) [TM T + a (TM r, — TiE r,)]r r4(rMr,.2B(l —a) +ar18r)(*)ihis is true given that:r r1>a>O, r>O, H>O,T T2Similarly, if XB as defined by (3-6) is less than or equal to II, then all that isneeded to have A fully specialize is for the optimal production division responseof B to the conditions assumed to be greater than or equal to the productiondivision required by the first order condition outlined above, i.e.,: XB*- XB° 0.Recall that more specialization for B means a larger production division, XB.Therefore, the condition where the optimal response of B allows fullspecialization by A is: exactly the same as (*) above.Thus far it has been proven that if XA > 0 in equilibrium then B fully specializesand if XB < H in equilibrium then A fully specializes. The only conditions thatviolate either of these inequalities entail at least one nation fully specializing bydefinition, as required.C184Appendix Fifteen: Proof that Specialization in the One-Stare Game is Greaterthan that in the Two-Stage GameHypothesis: Unless it is the case that both nations are fully specializing in the two-stage game,it will always be that the two-stage game involves less specialization than the one-stage game.Proof: Consider the following representative case where Nation B is fully specialized butNation A is not; A has XA = e > 0. Thus, first (and second) order conditions forNation A to maximize its welfare when B fully specializes must give XA = e:=0ax .XA€A X8-HIn the one-stage game this condition translates into the following equality:1-a e—————1a HaIn the two-stage game, the resulting equality is more complicated:1—a2 e(l+a)r =r [—______2a 2Ha+ je2(1— a)(1 + 3 a) + H2(1 — a2) — 2eH(1 — a)[(1 + a) + a(1 — a)]]2 Ha2185e can realistically range from 0, when A fully specializes, to H (1 - a), when Adecides for autarky production. Now consider a value for e that falls in betweenthese two limits:H (1- a)e = wherek> 1.kSubstituting this identity into the equality for the one-stage game gives:k-i 1—ak aSubstituting the identity into the equality for the two-stage game gives:— T (1 —a2)(k— 1)2a k+ (1 a) (1- a)(1 + 3 a) + k2(1 + a)2- 2k(1 + 2a - a2)]Now compare the multipliers on r2A under each game. The multiplier of the twostage game is larger than that of the one-stage game when:(1 —a)(k —1) ÷f(1 —a)(1 + 3a) +k2(1 ÷a)2 —2k(1 +2a —a2) >0This holds true for all 1 > a > 0 and k > 1 as the first product is larger thanzero by inspection and the terms in the square root bracket can be factored asfollows:186(k — 1)2 + a [2(k — 1)2] + a2 [(k + 3)(k — 1)] > 0.Thus, for a given level of specialization, e, for Nation A, the multiplier in thetwo-stage game is larger than that of the one-stage game. It can be seen that forthis example, a larger multiplier means a smaller e means more specialization(i.e., consider the equality of the one-stage game and the defmition of e).Therefore, less specialization means that the multiplier is less than that requiredby the equality condition (for the game in question) as defined above. Now if thesame parameters are used in both games, the ratio of r to r2A must be fixed sothe multiplier must be fixed. Consider a case where the multiplier satisfies theequality condition for the one-stage game. It is known that the multiplier willtherefore be smaller than that required for the condition of the two-stage game.Therefore, it necessarily follows that less specialization occurs in the two-stagegame than in the one-stage game (unless it is the case that the two-stage gameinvolves full specialization by both nations).ci187Appendix Sixteen: Concluding Proposition of Chapter ThreeProposition: The two-stage game can result in greater welfare than the one-stage game for onenation but not for both nations.Proof: There are a number of different aspects to consider in supporting this proposition.First, it must be shown that the two-stage game can result in greater welfare thanthe one-stage game for one nation. This must be shown for both the unrestrictedand the restricted final good stocks case. However, because it has been shownthat the two-stage game cannot be solved algebraically in general, only numericalexamples will be provided as support for this part of the proposition. Theexamples shown will not be pathological ones, but simply representative of asizeable set of examples.Second, it must be shown that the combined welfare of the two nations under thetwo-stage game is never greater than that under the one-stage game. This isaccomplished in two steps: 1) show that production specializations of nations arestrategic complements; and, 2) show that it is never the case that there is morespecialization in the two-stage game than in the one-stage game.188Example with unrestricted stocks:1 3Assume: H=10 a=— r=2 r=1 T1B=l r2—One-Stage Game Outcome: XA = 1.67 XB = 10p = 0.50 t = 10 YA = 1.90 B = 2.30Two -Stage Game Outcome: XA = 5.74 x8 = 8.43p = 0.91 t = = 2.00 Y2 = 2.04Nation A welfare under the two-stage gameis better than under the one-stage game.Example with restricted stocks:1 3Assume: H=10 a=— r=2 r=1 rlB=l r23=—also assume fish stocks are restricted at 3.8867 unitsOne-Stage Game Outcome: XA = 3.95 X = 6.22p = 0.837 t = 1.568 )‘4 = 2.0074 Y2 = 1.8793Two-Stage Game Outcome: XA = 5.00 XB = 6.00p = 1.000 t = 2.500 A = 2.0149 B = 1.8718Nation A “s welfare under the two -stage gameis better than under the one-stage game.The first part of the proposition has been proven. Also note that the examples areconsistent with the second part of the proposition.Before proving the difficult part of the second half of the proposition, the noncontroversial cases can be eliminated. The two games will result in the samecombined welfare for the two nations when either: i) the stocks are restricted toa level below S11 of the two-stage game; or ii) the nations are both fully189specializing in both games.It is therefore left to prove that combined welfare of the two-stage game is lessthan that of the one-stage game whenever final good stocks are unrestricted in thetwo-stage game sense, but not in the one-stage game sense (in general).Unless it is the case that in both games there is full specialization by both nations,it has been proven (in Appendix Fifteen) that more specialization occurs in theone-stage game than in the two-stage game. Therefore, if specializations arestrategic complements, then the final good stocks level that separates restrictedfrom unrestricted must be higher for the one-stage game than the two-stage game.If this is indeed the case, then the one-stage game must result in greater combinedwelfare than the two-stage game whenever final goods stocks are unrestricted inthe two-stage game sense.Thus, all that is left to prove the second half of the proposition is to show that thespecializations are strategic complements (under both games).For the one-stage game the following conditions hold:ax* r r= -[--(1 -a)÷_!a]<OaxB T1Aax* r r=- [—u (1 - a) ÷ — a] < 0aXA rlB r2B190The first condition shows that Nation A’s best response is to specialize further(i.e., decrease x1J whenever B increases its specialization; and the secondcondition shows that Nation B’s best response is also to specialize further whenA increases specialization. Therefore, specializations are strategic complements(i.e., both nations increase welfare when specializations increase under bestresponses).Proving that specializations are strategic complements in the two-stage game isa bit more complicated as no algebraic closed form solutions are available for bestresponses, in general. The proof is done indirectly as in Appendix Fifteen usinga representative algebraic example.First, generalizable but specific values are assumed for production specializations:XA = e > 0 and x = H z where z < 1. Now it is further assumed that e isNation A’s best response to B at this value of XB. A condition on the parametersis generated when this is assumed:8Y=0 x*=e>0 r =r Kax .X4C A 2B 2AA XBHZwhere: K is a multiplier like those of Appendix FifteenNow consider the effect on the multiplier, K, as z increases (i.e., XB increases),where K is a function of z and other parameters. From Appendix Fifteen, it isknown that a smaller e (i.e., increased specialization by A) requires a larger191multiplier. Some algebraic manipulation shows the following relationship betweenthe multiplier, K, and z:Thus, as B specializes more, z increases and K decreases. When K decreases, itis easier to meet the condition for XA = e. For a given set of parameters, themultiplier, K, is set by the parameters and not by the condition for XA = e. Fora given set of parameters, as XB increases, XA decreases because the condition onK is decreasing and now a condition which was once too high (and involved asmaller xJ can be satisfied by the multiplier.Therefore, specializations are also strategic complements in the two-stage game.This completes the proof of the second half of the proposition.0192


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