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Reflections on ‘social costs and benefits and the transfer pricing problem’
Journal of Public Economics 5 (1976) 353-359. 0 North-Holland
Publishing Company
REFLECTIONS ON ‘SOCIAL COSTS AND BENEFITS AND THE TRANSFER PRICING PROBLEM’ Theodore
GROVES
Graduate School of Management, Northwestern University, Evanston, IL 60201, U.S.A. Martin
LOEB
North Carolina State University at Raleigh, Raleigh, NC 27607, U.S.A. Received March 1975, revised version received July 1975 The informational incentive properties of a scheme proposed by Ronen to ensure efficient allocations in the presence of intertirm externalities are examined. While sending accurate information is a noncooperative (Nash) equilibrium in the game defined by Ronen’s scheme, it is not the only such equilibrium. Others such that all firms are better off and such that inefficient allocations result also exist. We show that under a scheme we proposed elsewhere, such possibilities are eliminated. Our scheme is similar to Ronen’s, but differs in two essential respects detailed in the paper.
1. Introduction In a recent issue of this Journal Ronen (1974) proposes a method to ensure efficient allocations in the presence of interfirm externalities. Ronen’s scheme, which is an adaptation of a transfer pricing scheme earlier proposed by Ronen and McKinney (1970) uses a central agent (the government) to collect information from the firms and to provide them with incentives to take socially optimal decisions by imposing taxes or subsidies. Ronen also claims that this scheme provides the firms with ‘incentives to supply correct information . . .’ [Ronen (1974, p. 80)]. In this note, we examine this claim more closely to point out some difficulties with Ronen’s scheme and to contrast it with a scheme we proposed elsewhere [Groves (1974) and Groves and Loeb (1975)J.
2. The Ronen scheme Ronen analyzes a two firm problem in which one firm engages in an activity with nonpecuniary diseconomies,l i.e. a ‘harmful activity’ that reduces the profits of the other firm. In order to motivate the firms to set the level of harmful activity at the socially optimal level, Ronen proposes that a government ‘When pecuniary diseconomies socially optimal allocations.
exist, maximizing joint profits will generally not lead to
3.54
T. Groves and M. Loeb, Social costs and benefits
agency (hereafter, the Center) collect information from the two firms, assess taxes on the damaging firm, and provide subsidies to the damaged firm. The Ronen scheme is illustrated in fig. 1 (our figures are similar to Ronen’s figures 3, 4, 5). The Center receives from the damaged firm (firm B) a marginal loss curve ML(B) reflecting how much of the harmful activity the damaged firm would be willing to accept at various subsidy rates. The Center then calculates an average (variable) loss curve for the damaged firm and communicates it to the damaging firm (firm A) which treats it as its tax schedule for various levels of the activity. Similarly, the Center receives from firm A a marginal gain curve MG(A) reflecting the level of the harmful activity it would choose to set at various tax rates. From this curve the Center calculates an average (variable) gain curve which it communicates to firm B as its subsidy schedule.
Value F
0
Level
Fig. 1. Nonoptimality
of Harmful Activity
of Nash equilibrium under Ronen’s scheme.
In Ronen’s scheme the Center’s role is limited to that of merely collecting and ‘transmitting information between the (firms) and paying the difference between the (damaged firm’s subsidies and the damaging firm’s taxes)’ [Ronen (1974, p. 75)]. Given the reported tax and subsidy schedules, the firms themselves decide the quantity of harmful activity under the stipulation that both must agree on the quantity. As Ronen shows, if both firms send their ‘true’ marginal loss and gain curves, ML*(B) and MG*(A) respectively, and are subsidized and taxed as indicated, they both would desire the socially optimal quantity, OD*, of the harmful activity. The subsidy to firm B equals the area OFED* and the tax on firm A equals the area OED*. Thus, the after-transfer profits of each firm is OEF, which also equals the Center’s deficit (subsidy less tax). Ronen further argues that ‘it can be easily shown that both (firms) would have incentives to supply correct information under this scheme’ [Ronen (1974,
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355
p. SO)]. Suppose, e.g., firm B sends its true marginal loss curve ML*(B) which then becomes firm A’s marginal tax schedule. Firm A then desires the quantity of harmful activity to be set at D* where MG*(A) = marginal tax = ML*(B). As firm A’s message will only affect the quantity that firm B desires (and not firm A’s tax schedule), firm A wishes to send a curve such that B will choose or accept the quantity OD*. Hence, firm A can do no better than by sending MC*(A). Similarly, ML*(B) is B’s best message given the supposition that firm A sends MG*(Ai In game theoretic language, the joint strategy consisting of both firms communicating the correct or true information [MG*(A), ML*(B)] is a Nash, or noncooperative, equilibrium. Serious objections, however, may be raised against the Ronen scheme since the truth-telling joint strategy is not generally the only Nash equilibrium that exists; others exist that leave both firms better off than does the ‘truthful’ Nash equilibrium. Thus, a reasonable presumption to suspect is that, if implemented, both firms would soon discover the Ronen scheme could be exploited. Furthermore, when such an exploitation occurs the resulting quantity of harmful activity may not be socially optimal. To see this, consider fig. 1 again. Suppose the damaging firm, A, overstates its (marginal) gains by sending JIG(A), and simultaneously the damaged firm, B, understates its (marginal) losses by sending ML(B). Firm A would want the quantity to be set where its true marginal gain, MG*(A), equals the reported marginal loss of firm B, ML(B). This quantity is seen to be OD. Firm B wishes the quantity to be set where its true marginal loss ML*(B) is equal to A’s reported marginal gain, MG(A). This quantity is also seen to be OB. Hence the nontruthful messages [MG(A), ML(B)] also form a Nash equilibrium. The aftertax profits of firm A are ODGF less its tax ODG, or OGF. Firm B receives a subsidy of ODHF from the Center and suffers a loss of OBH, thus leaving after transfer profits of OHF. Thus, by sending false information, firm A’s after transfer profits have increased by OGE and firm B’s profits have increased by FEH. The Center’s deficit has increased from OEF to OHGF. And, the level of harmful activity has increased from the socially optimal quantity OD* to the nonoptimal level OD. The result depends, of course, on limiting the Center’s role to exchanging information and assessing taxes and providing subsidies. In particular, the Center in Ronen’s scheme does not make the decision of how much of the harmful activity to allow or, equivalently, monitor the quantity chosen by the firms to see if it corresponds to the reported joint profit maximizing quantity. Ronen seems to have rejected centralization of the decision and indicates that his scheme’s ‘benefits include providing much of the firm’s autonomy’ [Ronen (1974, p. 79)]. However, since the Center has received both marginal loss and marginal gain curves, it would need no more information to make the decision. The benefits that would be foregone by this loss of autonomy seem obscure.
356
3. The Groves-Loeb
T. Groves and M. Loeb, Social costs and benefits
scheme
Elsewhere [Groves (1974) and Groves and Loeb (1975)] we have proposed a scheme, hereafter referred to as the G-L scheme, very similar to the Ronen scheme but with two crucial modifications. First, under the G-L scheme the Center has a decision-making role; it determines the level of harmful activity, choosing the quantity at which the reported marginal gain of the damaging firm equals the reported marginal loss of the damaged firm. The second modification is that an additional tax, depending only on the message of the other firm, is imposed on each firm. The purpose of these additional taxes is to guarantee that the Center’s net deficit (subsidies less taxes) is bounded above by zero. By centralizing the quantity decision, the G-L scheme has several desirable properties. Property I. Sending correct (‘truthful’) information is not only a Nash equilibrium, it is a dominant strategy equilibrium. That is, given any message of the other firm a firm is best off sending correct information. A formal proof of this property is in Groves and Loeb (1975) and Groves (1974); however, it is easy to indicate graphically why the property holds. Consider the damaging firm’s (firm A) problem. (The damaged firm’s problem is symmetric.) Suppose the damaged firm sends the Center the curve ML(B) which may or may not be the true marginal loss curve. In fig. 1 it is easy to see that if the damaging firm sends the marginal gain curve MG(A), which is different from its true marginal gain curve, MC*(A), the Center will choose the quantity Ofi and the damaging firm’s after-tax gross profits will be OGF less GKJ rather than OGF, which would have been realized if the firm had communicated MG*(A) instead. Although sending truthful information is a Nash equilibrium it is not the only Nash equilibrium. Property 2. Under a few regularity conditions, all Nash equilibria result in the socially optimal quantity of harmful activity - OD*. A formal proof of this property is in Loeb (1975); it follows almost immediately from property 1. Property 3. Although other Nash equilibria exist, under a few regularity conditions, the ‘truthful’ Nash equilibrium, that is, sending accurate information, is the unique dominant strategy equilibrium. A formal proof of this property is in Groves (1974); however, it can be seen intuitively by considering fig. 2. The strategy pair [MG(A), ML(B)] consisting of the curvilinear messages MG(A) and ML(B) is also a Nash equilibrium. However, if B were to send some other curve, say ML’(B), that intersects the
T. Groves and M. Loeb, Social costs and benefits
357
true marginal gain curve MG*(A) at a point different from E, then MG*(A) and not MG(A) would be A’s best response to A4L”(B). Thus, MG(A) is not dominant. Properties 1-3 of the G-L scheme all follow from centralizing the quantity decision and hence are not properties of Ronen’s scheme. In particular, all those Nash equilibria of Ronen’s scheme that lead to nonoptimal levels of the harmful activity and that are better for both firms than the truthful equilibrium are eliminated under the G-L scheme. However, without further modifications, this scheme would not be satisfactory since among the many Nash equilibria are also many nontruthful ones leaving both firms better off than at the truthful Nash equilibrium and leaving the Center with an even larger deficit to finance somehow.
ML*(B)
Fig. 2. Nontruthful
Nash equilibrium.
For example, in fig. 2, by sending the curvilinear messages MG(A) and ML(B) instead of the truthful messages MG*(A) and ML*(B), the damaging firm increases its after-transfer profits by the shaded region A + and the damaged firm increases its after-transfer profits by B+ . Also, the Center’s deficit has increased from OEF to OEF plus A + and B+. Since MG(A) and ML(B) are arbitrary except that they must intersect at E (to be a Nash equilibrium), areas A + and B+ may be made arbitrarily large. To eliminate the possibility of arbitrarily large deficits, the G-L scheme includes additional taxes. By assessing each firm an additional tax that does not depend on its own message, the incentive properties listed above are not affected and it is possible to guarantee the Center a nonnegative surplus. While there are a great many of such additional taxes, a particularly simple one is illustrated in fig. 3. Given any pair of messages [MG(A), ML(B)], the line PP’ will define two points, Q and R, of intersection with ML(B) and MG(A) respectively. The
358
T. Groves and M. Loeb, Social costs and benefits
additional tax of firm A is then calculated to be the area OPQ and the additional tax of firm B is area FPR. Since the additional tax of each firm depends only on the (arbitrary) line PP’ and the message of the other firm, the incentive properties of the G-L scheme are undisturbed by their addition to the scheme. Furthermore, since the deficit of the Center in the absence of these additional taxes would be area OHF, with these additional taxes the net surplus of the Center is the (positive) area QHR. Property 4. Under the G-L scheme, the Center’s deficit may be guaranteed to be nonpositive. Although the line PP’ is arbitrary, if it turns out to have been chosen so that the intersection points Q and R with the true marginal curves MC*(A) and Value F ,
Level Fig. 3. Groves-Loeb
scheme to guarantee no deficit.
ML*(B) are ‘far’ from the intersection point H of these curves, then it may be that these extra taxes will bankrupt one of the firms. Thus, to avoid this difficulty PP’ must be chosen to lie in some range that is possibly narrow and, in any event, not precisely known by the Center. For a fuller discussion of this issue and an interpretation of a procedure for instituting the scheme in such a way that no firm would be bankrupt, see Groves (1974) and Groves and Loeb (1975).
Now, although introducing these additional taxes does not eliminate as Nash equilibria nontruthful joint strategies such as [MG(A), ML(B)] of fig. 2, it does guarantee that the Center’s deficit is bounded above by zero. Thus, regardless of whatever type of collusive behavior the two firms may engage in or attempting to manipulate the scheme, the best they can jointly do2 is to maximise joint ‘This assuming the Center is immune to threats; we are indebted to a referee for raising this question.
T. Groves and M. Loeb, Social costs and benefits
359
profits and leave the Center with zero surplus. This does not mean, however, that any type of collusive behavior leaving both firms better off than at the ‘truthful’ dominant equilibrium also necessarily leads to the social optimum (the joint-profit maximizing level of the harmful activity, OD* of figs. 1 and 2). However, any collusive agreement leading to a nonsocial optimum is unstable in the sense that it is not a Nash equilibrium; this is not true of the Ronen scheme, as pointed out above. Furthermore, for the two-firm problem considered here, if the two firms collude it would be reasonable to expect them to achieve the joint-profit maximum (the social optimum), although if they would collude when faced with the G-L scheme, it also seems reasonable to assume they would collude in the absence of the scheme and fully internalize the externality, in which case there would be no need for any scheme in the first place. In contrast, under Ronen’s scheme, since collusion can lead to arbitrarily large profits for both firms, and consequently arbitrarily large deficits for the Center, his scheme provides very strong incentives for the firms to engage in such collusion. In cases of more than two firms, the collusion problem is more complex and we have not fully analyzed it as yet. However, the G-L scheme is fully generalizable to any number of firms and any number of externalities, and all four properties listed above still hold. the G-L scheme for treating interfirm Combining all these properties, externalities is seen to provide incentives for every firm to send ‘truthful’ information, regardless of the messages of the other firms, and furthermore is capable of guaranteeing that any collusive arrangement between the firms will not bankrupt the Center. In fact, as we point out in Groves (1974) and Groves and Loeb (1973, this scheme might be an attractive method for a group of firms affected by an externality to voluntarily agree among themselves to implement by hiring an agent to perform the role of the Center. Ronen’s scheme, since it guarantees a deficit for the Center, would be feasible only for a government with independent authority to finance its deficit in some manner.
References Groves, T., 1974, Information, incentives, and the internalization of production externalities, Working Paper No. 87, Center for Mathematical Studies in Economics and Management Science, Northwestern University; to appear in: S. Lin, ed., Theory and measurement of economic externalities (Academic Press, New York)a Groves, T. and M. Loeb, 1975, Incentives and public inputs, Journal of Public Economics 4, no. 3,21 l-226. Loeb, M., 1975, Coordination and informational incentive problems in the multidivisional firm, Unpublished doctoral dissertation (Northwestern University, Evanston). Ronen, J., 1974, Social costs and benefits and the transfer pricing problem, Journal of Public Economics 3, no. 1,71-82. Ronen, J. and G. McKinney III, 1970, Transfer pricing for divisional autonomy, Journal of Accounting Research, spring 99-112.
Publishing Company
REFLECTIONS ON ‘SOCIAL COSTS AND BENEFITS AND THE TRANSFER PRICING PROBLEM’ Theodore
GROVES
Graduate School of Management, Northwestern University, Evanston, IL 60201, U.S.A. Martin
LOEB
North Carolina State University at Raleigh, Raleigh, NC 27607, U.S.A. Received March 1975, revised version received July 1975 The informational incentive properties of a scheme proposed by Ronen to ensure efficient allocations in the presence of intertirm externalities are examined. While sending accurate information is a noncooperative (Nash) equilibrium in the game defined by Ronen’s scheme, it is not the only such equilibrium. Others such that all firms are better off and such that inefficient allocations result also exist. We show that under a scheme we proposed elsewhere, such possibilities are eliminated. Our scheme is similar to Ronen’s, but differs in two essential respects detailed in the paper.
1. Introduction In a recent issue of this Journal Ronen (1974) proposes a method to ensure efficient allocations in the presence of interfirm externalities. Ronen’s scheme, which is an adaptation of a transfer pricing scheme earlier proposed by Ronen and McKinney (1970) uses a central agent (the government) to collect information from the firms and to provide them with incentives to take socially optimal decisions by imposing taxes or subsidies. Ronen also claims that this scheme provides the firms with ‘incentives to supply correct information . . .’ [Ronen (1974, p. 80)]. In this note, we examine this claim more closely to point out some difficulties with Ronen’s scheme and to contrast it with a scheme we proposed elsewhere [Groves (1974) and Groves and Loeb (1975)J.
2. The Ronen scheme Ronen analyzes a two firm problem in which one firm engages in an activity with nonpecuniary diseconomies,l i.e. a ‘harmful activity’ that reduces the profits of the other firm. In order to motivate the firms to set the level of harmful activity at the socially optimal level, Ronen proposes that a government ‘When pecuniary diseconomies socially optimal allocations.
exist, maximizing joint profits will generally not lead to
3.54
T. Groves and M. Loeb, Social costs and benefits
agency (hereafter, the Center) collect information from the two firms, assess taxes on the damaging firm, and provide subsidies to the damaged firm. The Ronen scheme is illustrated in fig. 1 (our figures are similar to Ronen’s figures 3, 4, 5). The Center receives from the damaged firm (firm B) a marginal loss curve ML(B) reflecting how much of the harmful activity the damaged firm would be willing to accept at various subsidy rates. The Center then calculates an average (variable) loss curve for the damaged firm and communicates it to the damaging firm (firm A) which treats it as its tax schedule for various levels of the activity. Similarly, the Center receives from firm A a marginal gain curve MG(A) reflecting the level of the harmful activity it would choose to set at various tax rates. From this curve the Center calculates an average (variable) gain curve which it communicates to firm B as its subsidy schedule.
Value F
0
Level
Fig. 1. Nonoptimality
of Harmful Activity
of Nash equilibrium under Ronen’s scheme.
In Ronen’s scheme the Center’s role is limited to that of merely collecting and ‘transmitting information between the (firms) and paying the difference between the (damaged firm’s subsidies and the damaging firm’s taxes)’ [Ronen (1974, p. 75)]. Given the reported tax and subsidy schedules, the firms themselves decide the quantity of harmful activity under the stipulation that both must agree on the quantity. As Ronen shows, if both firms send their ‘true’ marginal loss and gain curves, ML*(B) and MG*(A) respectively, and are subsidized and taxed as indicated, they both would desire the socially optimal quantity, OD*, of the harmful activity. The subsidy to firm B equals the area OFED* and the tax on firm A equals the area OED*. Thus, the after-transfer profits of each firm is OEF, which also equals the Center’s deficit (subsidy less tax). Ronen further argues that ‘it can be easily shown that both (firms) would have incentives to supply correct information under this scheme’ [Ronen (1974,
T. Groves and M. Loeb, Social costs and benefits
355
p. SO)]. Suppose, e.g., firm B sends its true marginal loss curve ML*(B) which then becomes firm A’s marginal tax schedule. Firm A then desires the quantity of harmful activity to be set at D* where MG*(A) = marginal tax = ML*(B). As firm A’s message will only affect the quantity that firm B desires (and not firm A’s tax schedule), firm A wishes to send a curve such that B will choose or accept the quantity OD*. Hence, firm A can do no better than by sending MC*(A). Similarly, ML*(B) is B’s best message given the supposition that firm A sends MG*(Ai In game theoretic language, the joint strategy consisting of both firms communicating the correct or true information [MG*(A), ML*(B)] is a Nash, or noncooperative, equilibrium. Serious objections, however, may be raised against the Ronen scheme since the truth-telling joint strategy is not generally the only Nash equilibrium that exists; others exist that leave both firms better off than does the ‘truthful’ Nash equilibrium. Thus, a reasonable presumption to suspect is that, if implemented, both firms would soon discover the Ronen scheme could be exploited. Furthermore, when such an exploitation occurs the resulting quantity of harmful activity may not be socially optimal. To see this, consider fig. 1 again. Suppose the damaging firm, A, overstates its (marginal) gains by sending JIG(A), and simultaneously the damaged firm, B, understates its (marginal) losses by sending ML(B). Firm A would want the quantity to be set where its true marginal gain, MG*(A), equals the reported marginal loss of firm B, ML(B). This quantity is seen to be OD. Firm B wishes the quantity to be set where its true marginal loss ML*(B) is equal to A’s reported marginal gain, MG(A). This quantity is also seen to be OB. Hence the nontruthful messages [MG(A), ML(B)] also form a Nash equilibrium. The aftertax profits of firm A are ODGF less its tax ODG, or OGF. Firm B receives a subsidy of ODHF from the Center and suffers a loss of OBH, thus leaving after transfer profits of OHF. Thus, by sending false information, firm A’s after transfer profits have increased by OGE and firm B’s profits have increased by FEH. The Center’s deficit has increased from OEF to OHGF. And, the level of harmful activity has increased from the socially optimal quantity OD* to the nonoptimal level OD. The result depends, of course, on limiting the Center’s role to exchanging information and assessing taxes and providing subsidies. In particular, the Center in Ronen’s scheme does not make the decision of how much of the harmful activity to allow or, equivalently, monitor the quantity chosen by the firms to see if it corresponds to the reported joint profit maximizing quantity. Ronen seems to have rejected centralization of the decision and indicates that his scheme’s ‘benefits include providing much of the firm’s autonomy’ [Ronen (1974, p. 79)]. However, since the Center has received both marginal loss and marginal gain curves, it would need no more information to make the decision. The benefits that would be foregone by this loss of autonomy seem obscure.
356
3. The Groves-Loeb
T. Groves and M. Loeb, Social costs and benefits
scheme
Elsewhere [Groves (1974) and Groves and Loeb (1975)] we have proposed a scheme, hereafter referred to as the G-L scheme, very similar to the Ronen scheme but with two crucial modifications. First, under the G-L scheme the Center has a decision-making role; it determines the level of harmful activity, choosing the quantity at which the reported marginal gain of the damaging firm equals the reported marginal loss of the damaged firm. The second modification is that an additional tax, depending only on the message of the other firm, is imposed on each firm. The purpose of these additional taxes is to guarantee that the Center’s net deficit (subsidies less taxes) is bounded above by zero. By centralizing the quantity decision, the G-L scheme has several desirable properties. Property I. Sending correct (‘truthful’) information is not only a Nash equilibrium, it is a dominant strategy equilibrium. That is, given any message of the other firm a firm is best off sending correct information. A formal proof of this property is in Groves and Loeb (1975) and Groves (1974); however, it is easy to indicate graphically why the property holds. Consider the damaging firm’s (firm A) problem. (The damaged firm’s problem is symmetric.) Suppose the damaged firm sends the Center the curve ML(B) which may or may not be the true marginal loss curve. In fig. 1 it is easy to see that if the damaging firm sends the marginal gain curve MG(A), which is different from its true marginal gain curve, MC*(A), the Center will choose the quantity Ofi and the damaging firm’s after-tax gross profits will be OGF less GKJ rather than OGF, which would have been realized if the firm had communicated MG*(A) instead. Although sending truthful information is a Nash equilibrium it is not the only Nash equilibrium. Property 2. Under a few regularity conditions, all Nash equilibria result in the socially optimal quantity of harmful activity - OD*. A formal proof of this property is in Loeb (1975); it follows almost immediately from property 1. Property 3. Although other Nash equilibria exist, under a few regularity conditions, the ‘truthful’ Nash equilibrium, that is, sending accurate information, is the unique dominant strategy equilibrium. A formal proof of this property is in Groves (1974); however, it can be seen intuitively by considering fig. 2. The strategy pair [MG(A), ML(B)] consisting of the curvilinear messages MG(A) and ML(B) is also a Nash equilibrium. However, if B were to send some other curve, say ML’(B), that intersects the
T. Groves and M. Loeb, Social costs and benefits
357
true marginal gain curve MG*(A) at a point different from E, then MG*(A) and not MG(A) would be A’s best response to A4L”(B). Thus, MG(A) is not dominant. Properties 1-3 of the G-L scheme all follow from centralizing the quantity decision and hence are not properties of Ronen’s scheme. In particular, all those Nash equilibria of Ronen’s scheme that lead to nonoptimal levels of the harmful activity and that are better for both firms than the truthful equilibrium are eliminated under the G-L scheme. However, without further modifications, this scheme would not be satisfactory since among the many Nash equilibria are also many nontruthful ones leaving both firms better off than at the truthful Nash equilibrium and leaving the Center with an even larger deficit to finance somehow.
ML*(B)
Fig. 2. Nontruthful
Nash equilibrium.
For example, in fig. 2, by sending the curvilinear messages MG(A) and ML(B) instead of the truthful messages MG*(A) and ML*(B), the damaging firm increases its after-transfer profits by the shaded region A + and the damaged firm increases its after-transfer profits by B+ . Also, the Center’s deficit has increased from OEF to OEF plus A + and B+. Since MG(A) and ML(B) are arbitrary except that they must intersect at E (to be a Nash equilibrium), areas A + and B+ may be made arbitrarily large. To eliminate the possibility of arbitrarily large deficits, the G-L scheme includes additional taxes. By assessing each firm an additional tax that does not depend on its own message, the incentive properties listed above are not affected and it is possible to guarantee the Center a nonnegative surplus. While there are a great many of such additional taxes, a particularly simple one is illustrated in fig. 3. Given any pair of messages [MG(A), ML(B)], the line PP’ will define two points, Q and R, of intersection with ML(B) and MG(A) respectively. The
358
T. Groves and M. Loeb, Social costs and benefits
additional tax of firm A is then calculated to be the area OPQ and the additional tax of firm B is area FPR. Since the additional tax of each firm depends only on the (arbitrary) line PP’ and the message of the other firm, the incentive properties of the G-L scheme are undisturbed by their addition to the scheme. Furthermore, since the deficit of the Center in the absence of these additional taxes would be area OHF, with these additional taxes the net surplus of the Center is the (positive) area QHR. Property 4. Under the G-L scheme, the Center’s deficit may be guaranteed to be nonpositive. Although the line PP’ is arbitrary, if it turns out to have been chosen so that the intersection points Q and R with the true marginal curves MC*(A) and Value F ,
Level Fig. 3. Groves-Loeb
scheme to guarantee no deficit.
ML*(B) are ‘far’ from the intersection point H of these curves, then it may be that these extra taxes will bankrupt one of the firms. Thus, to avoid this difficulty PP’ must be chosen to lie in some range that is possibly narrow and, in any event, not precisely known by the Center. For a fuller discussion of this issue and an interpretation of a procedure for instituting the scheme in such a way that no firm would be bankrupt, see Groves (1974) and Groves and Loeb (1975).
Now, although introducing these additional taxes does not eliminate as Nash equilibria nontruthful joint strategies such as [MG(A), ML(B)] of fig. 2, it does guarantee that the Center’s deficit is bounded above by zero. Thus, regardless of whatever type of collusive behavior the two firms may engage in or attempting to manipulate the scheme, the best they can jointly do2 is to maximise joint ‘This assuming the Center is immune to threats; we are indebted to a referee for raising this question.
T. Groves and M. Loeb, Social costs and benefits
359
profits and leave the Center with zero surplus. This does not mean, however, that any type of collusive behavior leaving both firms better off than at the ‘truthful’ dominant equilibrium also necessarily leads to the social optimum (the joint-profit maximizing level of the harmful activity, OD* of figs. 1 and 2). However, any collusive agreement leading to a nonsocial optimum is unstable in the sense that it is not a Nash equilibrium; this is not true of the Ronen scheme, as pointed out above. Furthermore, for the two-firm problem considered here, if the two firms collude it would be reasonable to expect them to achieve the joint-profit maximum (the social optimum), although if they would collude when faced with the G-L scheme, it also seems reasonable to assume they would collude in the absence of the scheme and fully internalize the externality, in which case there would be no need for any scheme in the first place. In contrast, under Ronen’s scheme, since collusion can lead to arbitrarily large profits for both firms, and consequently arbitrarily large deficits for the Center, his scheme provides very strong incentives for the firms to engage in such collusion. In cases of more than two firms, the collusion problem is more complex and we have not fully analyzed it as yet. However, the G-L scheme is fully generalizable to any number of firms and any number of externalities, and all four properties listed above still hold. the G-L scheme for treating interfirm Combining all these properties, externalities is seen to provide incentives for every firm to send ‘truthful’ information, regardless of the messages of the other firms, and furthermore is capable of guaranteeing that any collusive arrangement between the firms will not bankrupt the Center. In fact, as we point out in Groves (1974) and Groves and Loeb (1973, this scheme might be an attractive method for a group of firms affected by an externality to voluntarily agree among themselves to implement by hiring an agent to perform the role of the Center. Ronen’s scheme, since it guarantees a deficit for the Center, would be feasible only for a government with independent authority to finance its deficit in some manner.
References Groves, T., 1974, Information, incentives, and the internalization of production externalities, Working Paper No. 87, Center for Mathematical Studies in Economics and Management Science, Northwestern University; to appear in: S. Lin, ed., Theory and measurement of economic externalities (Academic Press, New York)a Groves, T. and M. Loeb, 1975, Incentives and public inputs, Journal of Public Economics 4, no. 3,21 l-226. Loeb, M., 1975, Coordination and informational incentive problems in the multidivisional firm, Unpublished doctoral dissertation (Northwestern University, Evanston). Ronen, J., 1974, Social costs and benefits and the transfer pricing problem, Journal of Public Economics 3, no. 1,71-82. Ronen, J. and G. McKinney III, 1970, Transfer pricing for divisional autonomy, Journal of Accounting Research, spring 99-112.