On the Existence of Safety Equilibria Under Products ... - Springer Link

European Journal of Law and Economics, 5:153–165 (1998) © 1998 Kluwer Academic Publishers

Limited Liability and Imperfect Information—On the Existence of Safety Equilibria Under Products Liability Law ALFRED ENDRES Department of Economics, University of Hagen, Profilstra 8, 58084 Hagen, Germany ANDREAS LÜDEKE Department of Economics, University of Hagen, Profilstra 8, 58084 Hagen, Germany

Abstract In the literature on the economics of tort law the Cournot-Nash equilibrium concept has been undisputed. In particular, the existence of a Cournot-Nash safety equilibrium has been taken for granted, given the usual convexity assumptions. In this paper a simple model of strict liability with the defence of contributory negligence is considered. Liability is assumed to be limited. It is shown that for a certain range of liability limits no Cournot-Nash safety equilibrium in pure strategies exists. A mixed strategy equilibrium exists but it turns out to be suboptimal. Keywords: Tort Law, Product Liability, Product Safety, Mixed Strategy Equilibrium, Imperfect Information JEL Classification: K13

We consider a simple economic model of products liability law. A large number of identical firms offer a product which can cause damage to its consumers. Both, firms and consumers can reduce product risks by taking preventive measures. The accident prevention chosen by the firms is supposed not be known to the consumers. Under caveat emptor (no liability) asymmetric information about the safety of the products creates the problem of moral hazard on the side of the firm. The incentives of the firms to provide accident prevention are suboptimal.1 It is well known that under these circumstances a rule of strict liability with contributory negligence generates incentives for efficient accident prevention for both, firms and consumers (Shavell 1987, p. 52–4; Landes and Posner 1987, p. 79–80). However, this result requires the assumption of unrestricted liability of the firms. In this paper we deviate from this strong assumption. We introduce a complication common to many laws regulating torts: the limitation of liability. Another source of the limitation of liability is that the capital of firms, available for redemption, is limited for economic and legal reasons. Assuming this, it is shown that a Cournot-Nash care equilibrium (in pure strategies) does not exist if the amount limiting liability is chosen to be within a certain range. As a way out of this disequilibrium we derive a mixed strategy equilibrium. In addition, we

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show that a suboptimal no liability equilibrium is realised if the limitation of liability is too “low”. The paper is organised as follows. We start with deriving the Pareto optimal accident prevention and determine the equilibrium accident preventions and the product market equilibrium which result under caveat emptor and strict liability with contributory negligence. Then, we characterize the possible equilibria which arise if a liability limitation is introduced.

1. The model 1.1. Basic assumptions The magnitude of the damage is determined by investments in product safety (x) and the representative consumer’s care using the product (y). A representative firm’s safety cost is CF(x) and a consumer’s care cost is CC(y). The probability of an accident to occur is assumed to be constant at p, the damage in the case of an accident is D(x, y).2 The safety, care and damage cost are costs of one product unit. Safety and damage costs of the product shall not depend on the total quantity produced. Then, the Pareto optimal safety (x*) and care (y*) levels are simply defined by minimizing total expected cost of one product unit, i.e., by C 5 CF~x! 1 CC~y! 1 pD~x, y! 5 min.

(1)

x* and y* simultaneously solve ]CF~x! ]x

]pD~x, y!

52

]x

and

]CC~y! ]y

]pD~x, y! . ]y

52

(2)

Let P(s) be the inverse demand function, where s is the product market output. The inverse demand is a function of the marginal utility of the product minus care costs and damages consumers have to bear in case of an accident: P~s! 5 u8~s! 2 CC~y! 2 fpD~x, y!, with f [ {0, 1%.

(3)

f denotes the fraction of damages consumers bear in case of an accident. Competition in the product market is supposed to be perfect such that the market price equals per unit costs. Then, the Pareto optimal quantity s* of the product is obviously determined by u8~s! 2 CC~y*! 5 CF~x*! 1 pD~x*, y*!

(4)

given the equilibrium safety and care tupel x*, y*. To simplify the analysis (and the formal terms given below, in particular) marginal production costs are assumed to be constant and normalized to zero.


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1.2. Caveat emptor If the consumers bear the damage (f51) the firm’s safety related costs consist of safety costs, only, CF~x!.

(5)

Accordingly, consumer’s costs are CC~y! 1 pD~x, y!.

(6)

To simplify, we assume that consumers cannot sue firms for providing a faulty safety level. This may be so because consumers are also unable to detect the safety level after purchase. Since providing safety is costly for the firms and the consumers have no information about the safety level provided, the firms reduce safety to the lowest possible level, x 5 0. With x 5 0 the consumer’s cost function changes to CC~y! 1 pD~x 5 0, y!.

(7)

Consumers with rational expectations do not believe that firms provide products with higher than minimum safety. Their best answer is to choose a care level y° which minimizes (7). (It follows from footnote 1 that y° . y*.) The equilibrium safety and care tupel under no liability (x 5 0, y°) does not minimize the total expected cost of one unit (see (2)). Consequently, for the quantity traded in equilibrium s° , s* holds (see (4)). Thus, under a rule of no liability imperfect information of consumers results in suboptimal accident prevention and a suboptimal quantity. In the next section we show a liability rule which results in Pareto optimal safety, care and quantity choices.

1.3. Strict and unlimited liability We consider the rule of strict liability with contributory negligence. We assume that the firm compensates consumers perfectly if these fulfil the legal care standard. The courts are assumed to be able to find out about the care chosen by the consumers. So violation of the due care standards is detected in this model. In this case f 5 0 holds. The legal standard for the consumer’s care level is supposed to be equal to the Pareto optimal level, y*, 2 a textbook assumption in the economics of tort law.3 Accordingly, consumer’s costs are CC~y! CC~y! 1 pD~x, y!

for y $ y*

(8a)

for y , y*

(8b)

The firm’s costs are


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for y $ y*

(9a)

for y , y*

CF~x!

(9b)

Choosing the Pareto optimal care level y* is a dominant strategy for the consumer. Given this strategy the firm minimizes its costs by choosing the Pareto optimal safety level x*. In this simple model the Pareto optimal values x*, y*, represent the unique, stable safety/ care equilibrium.4 Given the Pareto optimal safety and care, the quantity s° traded in equilibrium is determined by P~s! 5 CC~y*! 1 pD~x*, y*!.

(10)

With (3) and f 5 0, this condition is identical to the Pareto condition (4), i. e., s° 5 s* holds. In the next section, a complication common to many laws regulating torts is introduced to the textbook case sketched above: the limitation of liability. Limitations of liability are a most common feature in tort law. E.g., liability is limited in German environmental liability law, biotechnology law and nuclear law. Beyond the regulations of safety and liability law the value of a firm’s assets creates an upper boundary to liability.

2. The limitation of liability 2.1. The modified cost functions We continue to consider strict liability with contributory negligence. Now liability shall be ¯ , so the maximum expected compensation payment for the firm limited to an amount of D ¯ is pD 5 const. Given this limitation, the cost situation of a firm is changed from (9a), (9b) above, to ¯ CF~x! 1 pD

for x # x# and y $ y*

(11a)

CF~x! 1 pD~x, y!

for x $ x# and y $ y*

(11b)

CF~x!

for y , y*

(11c)

¯ 5 D(x, y). So x# is a critical safety level, above which the In (11a), (11b), x# is defined by D limitation of liability ceases to be binding. The representative consumer’s costs are CC~y!

for x $ x# and y $ y*

¯ CC~y! 1 pD~x, y! 2 pD


(12a)

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for y , y*

(12c)

Before dealing with the allocative consequences of different levels of liability limitation a few things should be observed regarding the cost functions (11) and (12): 1. Given the consumer keeps the legal standard (y $ y*), the firm’s cost function consists of two parts. Given the assumptions of footnote 2, for “low” levels of safety (x # x# ), the firm’s total expected costs are the cost of safety, CF(cx), plus the maximum ¯ . This is represented by (11a). For higher safety expected compensation payment, pD levels (x $ x# ) the firm’s total expected costs are the cost of safety measures, CF(x), plus expected damage, pD(x, y). This is represented by (11b). So the firm’s cost function has two local minima: (11a) is minimized by x 5 0 and let (11b) be minimized by x 5 g(y). This two part cost function is illustrated in Fig. 1. It is shown as the lower boundary ¯ and CF(x) 1 pD(x, yˆ). In the figure we take of the structure consisting of CF(x) 1 pD the consumer’s choice of some care level yˆ as a parameter. 2. Regarding the consumer’s costs it is obvious that (12a) is minimized by y* since CC(y) is monotonically increasing in y. Let (12b) be minimized by y 5 f(x). 3. Equations (11c) and (12c) will be disregarded in the following equilibrium analysis. The reason is that they both hold for y , y*. Ignoring the due care standard, however, is a strictly dominated strategy for the consumer, i.e., it is not chosen in equilibrium.

Figure 1.


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Given the relevant expected cost functions, the existence of a safety/care equilibrium in pure strategies and (in case it exists) the values of the equilibrium safety/care level depend ¯. upon the size of the liability limitation D Below, three ranges of liability limitation sizes are identified. The sizes of limitations falling into these categories are called “low”, “high” and “intermediate”, respectively. It is shown that in the case of a “low” limit a Cournot-Nash equilibrium exists, but is not Pareto optimal. In the case of a “high” liability limit the safety/care equilibrium is Pareto optimal. In the “intermediate” case no Cournot-Nash equilibrium in pure strategies exists.

2.2.1. The “low” limit case. Define y° 5 f(x 5 0) to be arg.min. CC(y) 1 pD(x 5 0, y) ¯ and x° 5 g(y°) to be arg.min. CF(x) 1 pD(x, y°). (It follows from footnote 2 that 2 pD y° . y* and x° , x*.) Then, x 5 0, y° is the equilibrium safety/care tupel if the liability limit meets ¯ , C ~x°! 1 pD~x°, y°!. pD F

(13)

¯ is said to be “low”. With (13) holding the liability limit D ¯ . The In the equilibrium situation x 5 0, y° the expected cost the firm faces is pD # consumer’s cost is CC(y°) 1 pD(x 5 0, y°) 2 pD. From this situation neither of the parties wants to deviate: y° is defined to be consumer’s best “answer” to x 5 0. Given y 5 y°, the firm faces a cost function with two local minima: For 0 # x # x# the cost function is (11a) with its minimum at x 5 0, for x $ x# the cost is (11b) with its minimum at x°. The cost at x 5 ¯ , the cost at x° is CF(x°) 1 pD(x°, y°).5 0 is pD So given condition (13), x 5 0 is the minimum minimorum. Thus, if the liability limit is “low” the situation is equivalent to the case of no liability. The safety, care, and quantity levels chosen in equilibrium are suboptimal.

2.2.2. The “high” limit case. On the other hand, if the limitation of liability meets ¯ . C ~x*! 1 pD~x*, y*!, pD F

(14)

the Pareto optimal quantities x*, y* are the Cournot-Nash equilibrium. ¯ is said to be “high”. Here, the limitation of With (14) holding the liability limit D liability is not binding in equilibrium. Therefore, the text book result prevails: A strict liability rule with a defence of contributory negligence provides for efficient equilibria if the due care standard is set at its Pareto optimal level. Thus, if the liability limit is “high” the situation equals the unconstrained liability case resulting in a equilibrium with Pareto optimal safety, care, and quantity levels.


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2.2.3. The “intermediate” limit case. Because y° . y* (as has been argued above) and dD/dy , 0 it follows that D(x, y°) , D(x, y*) for any level of x. Since CF(x) is monotonly increasing in x, and x° , x* (as has been argued above), it follows that CF(x°) , CF(x*) holds. Thus, CF(x°) 1 pD(x°, y°) , CF(x*) 1 pD(x*, y*) prevails, with x° 5 arg.min. CF(x) 1 pD(x, y°) and x* 5 arg.min. CF(x) 1 pD(x, y*). So having considered the implications of ¯, ¯ , C ~x°! 1 pD~x°, y°! and C ~x*! 1 pD~x*, y*! , D pD F F ¯ might just as well fall in the two preceding paragraphs, we now have to face it that pD between CF(x°) 1 pD(x°, y°) and CF(x*) 1 pD(x*, y*). It is in this intermediate case, ¯ , C ~x*! 1 pD~x*, y*!, CF~x°! 1 pD~x°, y°! , pD F

(15)

no Cournot-Nash safety/care equilibrium in pure strategies exists. To see this consider an initial situation to be characterized by the Pareto optimal safety/care tupel x*, y*. It follows from the right hand side unequality of (15) that the firm decides to deviate from this situation by switching from x* to x 5 0, using the shelter of the liability limitation. Given x 5 0, the consumer adjusts by switching from y* to y° 5 f(x 5 0), according to condition (12b). Given y°, however, the firm improves its situation by expanding safety from x 5 0 to x° according to the left hand unequality in (15). Given that, the consumer reduces costs by reducing care from y° to y¯, with y¯ 5 arg.min. CC(y) ¯. 1 pD(x°, y) 2 pD So starting from a situation x 5 0, y°, a “process”6 of increasing x and decreasing y is induced. With dD/d(x, y) , 0, in this process, the firm’s cost function CF(x) 1 pD(x, y) is shifted upwards since y goes down. This process does not go through until the unique optimum x*, y* is attained. To the contrary, it is interrupted as soon as a situation y˜, x˜ 5 g(y˜) is attained for which ¯ CF~x! 1 pD~x, y! 5 pD

(16)

holds. As soon as the consumer’s care level drops infinitesimally below y˜, say to y˜ 2 e, in the process of his reduction of care, (16) turns into an inequality with the higher value on its left hand side. Therefore, instead of following the consumer’s reduction of care by expanding its safety level, the firm switches to exercising no safety activity at all, x 5 0, ¯ . Accordingly, the consumer adjusts by switching from y˜ incurring an expected cost of pD 2 e to y°. So instead of reaching an equilibrium, the “process” of mutual Cournot-Nash adjustments leads to a situation which has been shown not to be an equilibrium situation at the beginning of our “pseudodisequilibrium dynamics”.7


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The argument presented above is illustrated in Fig. 2 by following through from situation P0 to P4. An alternative illustration can be given in terms of the Cournot-Nash reaction curves of the involved parties (see figure 3). Without a limitation of liability the two reaction curves intersect in the point PCN, the unique Pareto optimal Cournot-Nash safety/care equilibrium x*, y*. However, with a limitation of liability which happens to be “intermediate” in the sense defined above, the reaction function of the potential injurer (the firm) x 5 g(y) is identical to the unconstrained one only in the range of y $ y˜. In y˜, the reaction function is discontinuous. The “jump” takes the reaction function right to the ordinate, showing that the best adjustment to any level of y , y˜ is to choose x 5 0. So the firm’s reaction curve, as modified by the “intermediate” limitation of liability does not intersect the consumer’s reaction curve and no Cournot-Nash equilibrium in pure strategies exists. As was shown the quantity traded is determined given the equilibrium safety/care tupel. So, if no equilibrium in pure strategies exists no equilibrium quantity can be found either. However, as we are going to show below, there is a mixed strategy (safety/care) equilibrium. Given this mixed strategy equilibrium an equilibrium quantity can be determined.

2.3. The mixed equilibrium In the case of a low and a high liability limitation according to (13) and (14), respectively, the players (firm, consumer) choose pure strategies. In the case of a low liability limitation the equilibrium safety/care levels x 5 0, y° are chosen with certainty. The same has been

Figure 2.


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Figure 3.

shown to hold for the Pareto optimal equilibrium safety and care levels x*, y* in the case of a high liability limitation. However, in the intermediate case (15) the firm chooses both actions x 5 0, x* and the consumer both actions y*, y° with positive probability, playing a mixed strategy. The firm would select x 5 0 with probability PF and x* with probability 1 2 PF. The consumer would choose y* with probability PC and y° with probability 1 2 PC. The probabilities P*F and P*C define a mixed equilibrium of the game, if the firm (consumer) minimizes its expected costs by choosing P*F(P*C) given P*C(P*F). With the payoffs from the matrix (see Fig.4), the firm minimizes # 1 ~1 2 P !P ~C ~x*! 1 pD~x*, y*!! 1 ~1 2 P !~1 2 P !~C ~x*! PFpD F C F F C F 1 pD~x*, y°!! and the consumer minimizes ¯ ! 1 P ~1 2 P !~C ~y°! 1 pD~0, y°! 2 p D # PFPC~CC~y*! 1 pD~0, y*! 2 pD F C C V ! 1 ~1 2 PF !PC CC ~y*! 1 ~1 2 PF! ~1 2 Pc! CC ~y°! It follows that the mixed strategies P*F 5

CC~y°! 2 CC~y*! pD~0, y*! 2 pD~0, y°)

and


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Figure 4.

P*C 5

¯ 2 C ~x*! 2 pD~x*, y°! pD F pD~x*, y*! 2 pD~x*, y°!

constitute an equilibrium. Interestingly, we see that the equilibrium strategy of the firm (consumer) depends on the costs of the other player. This leads to counterintuitive comparative static results. This has been demonstrated by Tsebelis (1989, 1991) and Wittman (1985, 1993). It should be noted that requiring one party to be aware of the other party’s costs is quite restrictive. For ¯ increases but (15) continues to hold the mixed strategy of example, if the liability limit D the consumer changes but not the mixed strategy of the firm. So, increasing the expected liability of the firm within the “intermediate” range does not has any influence on the accident prevention of the firm. In the context of crime prevention it was also shown by Tsebelis (1989) that higher penalties or sanctions have no additional deterrent effect on criminal behavior if the equilibrium is in mixed strategies. The expected social costs of the mixed equilibrium are higher than in the socially optimal situation, because the Pareto optimal safety and care levels x*, y* are realized with a probability (1 2 P*F)P*C , 1. Given the mixed strategy equilibrium P*F, P*C we can calculate an expected value E(s) for the quantity traded. The four safety/care combinations given in the matrix above determine four quantity levels. The sum of the four quantity levels weighted with the probabilities for the realisation of the four quantity levels gives the expected equilibrium quantity. Because the Pareto optimal safety/care level is reached with a probability smaller than one it is obvious that for the expected quantity E(s) , s* holds.


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How about the real world relevance of this mixed strategy equilibrium (and others)? There has been considerable criticism in the literature picking at the fact that it does not seem to be empirically plausible to have decision makers randomly choosing their actions (see, e. g., Eichberger (1993), p. 144–5).8 According to Harsanyi (1973), however, there is an interpretation for mixed strategies that does not use the idea of agents’ actual mixing behavior.9 Instead, it requires the introduction of an additional element of asymmetric information into the model. Besides imperfect information about the safety level, information about the payoffs is considered to be incomplete. In this modified setting each player chooses a pure strategy, but for the rival player it seems like mixing. Incomplete information about the opponent’s payoffs creates strategic uncertainty of a player about the choice of the other player. A mixed strategy can be seen as a formalisation of this uncertainty by means of a probability distribution over pure strategies. For example, we suppose that the firm (consumer) has “some minor private information” about his own cost of safety (care) if the safety (care) level x*(y*) is chosen: To incorporate private information into the payoff matrix (Fig. 4) the payoffs of the firm in cells III and IV are disturbed by an small amount of safety costs a and the payoffs of the consumer in cells I and III by an small amount of care costs b. a and b are random variables. If we assume that the probability distribution of a and b is common knowledge, but the actual value of a(b) in the game is only known to the firm (consumer), we get an incomplete information version of our simple economic products liability model. In the Bayes-Nash equilibrium of the game the players play pure strategies which are best responses to each other given the firm knows a and the consumer knows b. Because of the randomness of the distribution of a and b the play of the rival player will be perceived as mixing. According to Harsanyi (1973), it can be shown that the Bayes-Nash equilibrium of this game approximates the mixed strategy equilibrium of the game with perfect information (a 5 b 5 0), if the uncertainty becomes very small. So with an appropriate form of incomplete information about small fluctuations in the payoffs of the players we get a mixed strategy equilibrium without randomizing players.

3. Conclusion The incentives to take care provided by tort law have been widely analysed in the recent law and economics literature. In this paper the implications of limited liability have been considered in a products liability context. It has been shown that if the liability limit is below a certain amount the potential injurer (the firm) chooses no care (safety) at all in equilibrium. The potential victim’s (the consumer’s) care level is adjusted to this dismal situation. If the liability limit is above a certain limit (higher than the afore-mentioned one) the equilibrium care level of the potential injurer and the potential victim turn out to be the Pareto optimal ones. However, if the liability limit happens to be between the two afore-mentioned boundaries, no equilibrium exists in pure strategies. Only a mixed strategy equilibrium exists, which turns out to be suboptimal. It is important to note that these


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results have not been produced by deviating from the standard convexity assumptions. To the contrary, they have been shown to be the consequence of a common element of tort law itself: the limitation of liability. Regarding the mixed strategy equilibrium we raised some scepticism about the practical relevance of randomizing behavior. Therefore we have given an alternative interpretation of mixed strategies, which is based on lack of information about players’ payoffs. The mixed equilibrium has been shown to be the solution of a game with incomplete information, when this incompleteness becomes very small.

Acknowledgments The authors gratefully acknowledge grant No. En 192/3-1 by the German Research Foundation (DFG). They are indebted to two anonymous referees for their helpful comments on an earlier draft of this paper.

Notes 1. For an overview about market failure in product markets because of imperfect consumer information see Tirole (1988, Ch. 2). There is no reputation mechanism in the model considered below. 2. The standard assumptions in the literature on the form of these functions are dCF/dx . 0, d2CF/dx2 $ 0, dCC/dy . 0, d2CC/dy2 $ 0, dD/dx, dD/dy , 0, d2D/dx2,d2D/dy2 . 0, d2D/dxdy . 0. See, e.g., Brown’s seminal paper (1973). The functions used in this paper are supposed to show these properties. 3. See Stephen (1994) for a discussion of different explanations regarding the genesis of due care standards applied in the judiciary. The author refutes the “probabilistic interpretation” commonly accepted in the literature. Instead, he offers an explanation using Shackle’s criteria for decision making under uncertainty. 4. See, e.g., Landes and Posner (1987), Shavell (1987), and for the explicit use of a game-theoretic approach Baird, Gertner, and Picker (1994, ch. 1). 5. It is supposed that x° . x¯. Given that, the limitation of liability is not binding in x° i.e., the firm’s expected ¯ ). If x° , x¯ holds, the result of the main text that the fim chooses compensation payment is pD (and not pD x 5 0 is trivial, since the minimum at x 5 0 is unique. 6. Of course the term “process” is not be taken literally in this context since the model presented above is not a dynamic one. Instead, the argument in the text is carried on “by pseudodisequilibrium dynamics”. (See, e.g., Russell and Wilkinson (1979, p. 282). 7. Problems of the existence of care equilibria in pure strategies have also been discussed in Endres and Querner (1995). However, contrary to the paper at hand, in the former paper accidents are between strangers, there is no limitation of liability, and the due care standard is set at a suboptimal level. 8. Even if we think of randomizing as realistic behavior of the agents in some cases, the mixed strategy equilibrium seems to be weak. Because in a Nash-equilibrium in mixed strategies a player is indifferent with regard to his strategy choice there is an incentive problem to choose the Nash-equilibrium strategy. Given the equilibrium strategy of the opponent any mixture of the two pure strategies of the player is a best reply. The mixed strategy is only chosen to induce the opponent also to choose his/her mixed strategy. Recently, Cheng and Zhu (1995) have shown that these difficulties with mixed strategies depend on the assumption that the agents behave according to expected utility theory. As shown by Holler (1990, 1993), because of the indifference with regard to the strategy choice there may be a strong incentive for the players to deviate and to choose maximin-strategies. See Endres (1991) for the use of maximin strategies under tort law as a means to avoid the problems of the non existence of a pure strategy equilibrium.


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9. See Fudenberg and Tirole (1992), p. 230–4, for a detailed textbook presentation of this argument.

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