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Preference revelation and monopsony
Journal
of Public
Economics
20 (1983) 357-372.
PREFERENCE
North-Holland
REVELATION
Publishing
AND
Company
MONOPSONY
G. FANE University
of Victoria and Australian
National
University,
Canberra, A.C.?: 2600, Australia
E. SIEPER Australian
Received
National
February
University,
Canberra, A.C.7: 2600, Australia
1981, revised version
received
November
1981
(1) Johnson demonstrated the formal equivalence between the free-rider problem and monopsony. (2) Vickrey’s method for eliminating the inefliciencies resulting from monopsony (monopoly) works by converting simple monopsonists (monopolists) into perfectly discriminating monopsonists (monopolists). (3) We show that three recently proposed solutions to the free-rider problem have the same rationale, i.e. they use ‘Clarke taxes’ to convert Johnson’s simple monopsonists into perfectly discriminating monopsonists. The three mechanisms are those of: (i) Clarke and Groves; (ii) Groves and Ledyard; and (iii) Arrow. The relationships between these mechanisms, and between each of them and monopsony, are analyzed diagrammatically.
1. Introduction The purpose of this paper is to synthesize some recent contributions to the literature on preference revelation and the free-rider problem. Our basic point is that all the preference revealing mechanisms which we discuss are best understood in terms of the theory of monopsony, and all can be analyzed using the same simple geometric approach. Johnson (1977) noted the formal equivalence between the free-rider problem and monopsony: when a consumer-taxpayer is invited to state his willingness-to-pay for a public good and charged accordingly he is in the position of a monopsonist; his self-interest involves ‘free-riding’, i.e. the understatement of his true willingness-to-pay for the public good so as to reduce the average price which he pays. This leads to an inefficiently low level of output of the public good for just the same reasons that simple monopsony leads to inefficiently low output in markets for private goods. In the next section of this paper we explain this idea of Johnson’s in more detail, and in section 3 we show that the demand revealing process [i.e. the mechanism due to Clarke (1971), Groves (1973) and Groves and Loeb (1975)], henceforth the DRP, succeeds in inducing truthful preference revelation because it converts the simple monopsonists of Johnson’s problem 0047-2727/83/$03.00
0
Elsevier Science Publishers
B.V. (North-Holland)
358
G. Fane and E. Sieper, Preference revelation and monopsony
into perfectly discriminating monopsonists. It is of course well known that perfectly discriminating monopsony avoids the inefficiencies which result from simple monopsony. The formal equivalence between the DRP and the method proposed by Vickrey (1961) for overcoming the inefficiencies caused by monopsonies and monopolies is well known.’ In essence Vickrey’s method is to convert simple monopolists into perfectly discriminating monopolists by paying to each the area under his demand curve, and to convert simple monopsonists into perfectly discriminating monopsonists by charging each the area under his supply curve. The role of the so-called ‘Clarke taxes’ in the DRP is to ensure that Johnson’s representative monopsonist is charged the area under his supply curve, rather than the revenue rectangle defined by the chosen point on this curve. Given Johnson’s demonstration of the formal equivalence between the free-rider problem and monopsony it is not surprising that Vickrey’s method should also be the basis for the mechanisms for overcoming the free-rider problem. The existence of a relationship between the DRP and discriminating monopoly was noted by Groves (1973). In his model the resource manager of a conglomerate-type organization in which there are production externalities between the constituent sub-units attempts to set prices for each activity and for each sub-unit so as to maximize profits. After defining the ‘prices’ which yield optimal incentives, Groves notes that: While it is perhaps suggestive to represent the own profit incentive structure in terms of prices charged for resources allocated, it should be observed that these prices are in general not only different for the different enterprises, but also that the rules (functions) by which they are calculated are different for each enterprise. This non-anonymity property is unconventional, but has some parallel to the situation of a discriminating monopolist [Groves (1973, p. 629) emphasis added]. In sections 4 and 5 we show that the Groves-Ledyard optimal mechanism [Groves and Ledyard (1977, 1980)] henceforth the GLOM, and the demand revealing game, DRG [Arrow (1977)] also implicitly use Clarke taxes to convert simple monopsonists into perfectly discriminating monopsonists. Interpreting the GLOM and the DRG in terms of Clarke taxes allows us to exposit these mechanisms in terms of our basic geometric analysis of monopsony, and also helps to clarify the apparently disparate relationships between these mechanisms, the DRP and the contributions of Johnson and Vickrey. Finally, our analysis reveals why a necessary assumption for truthful revelation of preferences under the DRP is that there are no income effects in ‘See Tideman and Tullock (1976, p. 1146).
G. Fane and E. Sieper, Preference revelation and monopsony
the demand for the public good, and why this assumption for truthful revelation of preferences under the GLOM.
359
is not necessary
2. The free-rider problem as an instance of simple monopsony We begin this section with Johnson’s demonstration of the formal equivalence between the free-rider problem and simple monopsony. For simplicity, consider an economy with one private good, X, one public good, I: and just two consumer-taxpayers. Their true marginal evaluations of the public good are d,(y) and d2(y). Throughout sections 2, 3 and 5 we assume that each individual’s marginal evaluation of Y is independent of his level of consumption of the private good.2 The marginal cost of the public good is a constant, c. The government determines the output of the public good by asking each individual to announce his marginal evaluation and selecting the output level, Y; at which the sum of the announced marginal evaluations is equal to the marginal cost of the public good:
W)+&(y”)=c,
(1)
where &y”) denotes individual i’s announced marginal evaluation. The output of the public good is financed by requiring each individual to pay an average price for all units of Y equal to his announced marginal evaluation of the last unit produced. This method of providing public goods is usually referred to as the Lindahl mechanism. It is clear that the revenue raised is just sufficient to finance the output of the public good, leaving the government with a balanced budget. If each individual announces his true marginal evaluation schedule then the mechanism also achieves a Pareto efficient allocation, since eq. (1) then becomes the familiar efficiency condition c MRS= MRT. However, as Johnson pointed out, each individual has an incentive to announce a false marginal evaluation schedule because each is in the position of a monopsonist. This point is clear from fig. 1, in which amounts of the public good are measured on the abscissa and per unit evaluations and costs of the public good are measured on the ordinate. The vertical distance between 0, and 0, is c. With 0, as origin the curve labelled d,(y) represents individual l’s true marginal evaluation of Y; the curve labelled c -J2(y) can be interpreted in two ways: with 0, as origin and reading downwards it represents individual 2’s announced marginal evaluation of x with 0, as origin and reading upwards it represents the average price which individual 1 must pay per unit of Y as a function of the amount of Y.which individual 1 obtains, given the government’s rules. If individual 1 were to announce his true marginal evaluation schedule, d,(y), the government would select output ‘It is assumed that i’s utility function can be written in the form: ui( y,x,) = ui(y)+xi, true marginal evaluation of I: in terms of X, is given by: di( y) = vx y).
so that i’s
360
G. Fane and E. Sieper, Preference revelation and monopsony
MC,(Y)
c-
4
1,”
?
Fig. 1
level y*. However, given the schedule J2(y), individual 1 has an incentive free ride: the marginal cost of Y to individual 1, MC,(y), is given by:
to
This schedule, which is the marginal schedule derived from l’s average price schedule, is drawn in fig. 1. By announcing a false schedule, such as d;(y), individual 1 maximizes utility, since this schedule achieves the output level, p, at which l’s true marginal evaluation of Y is equal to the private marginal cost of Y to 1. By acting in this way 1 is exercising his monopsony power. For the purposes of comparing the DRP, the GLOM and the DRG with each other, and for showing how each of these mechanisms attempts to overcome the problem of monopsony in the Lindahl mechanism, it is helpful to redefine the Lindahl mechanism in net terms rather than gross terms: whereas di(y) and &(y) denote individual i’s true and announced marginal evaluation schedules for the public good, we now define ni( y) and yii(y) as his true and announced net marginal evaluations of the public good given that the government requires individual i to pay a share xi in the cost of providing the public good. These shares are determined so that xiai= 1. The true net marginal evaluations are:3 3An additional
advantage
of working
in net terms is that it clearly
reveals that there is no loss
G. Fane and E. Sieper, Preference revelation and monopson)
ni(Y)=di(Y)-aic, The announced
i=
net marginal
42.
evaluations i=
Ai(y)=~i(y)-_cl,C,
are:
1,2.
The government selects the output marginal evaluations sum to zero: &(Jq+ti,(y~=O.
361
level, 9, at which
the announced
net
(1’)
Individual 1 is required to make a side-payment of A,(j) to individual 2 for every unit of Y produced. Therefore the total payment by 1, per unit of Y is:
Similarly, individual 2 can be thought of as making a side-payment of &(y”), per unit of Y produced, to individual 1. Since &(y’)= -izi(y?, these two methods of defining the side-payment are equivalent definitions of a single transfer. The gross cost of Y to individual 2 is still a,(y7) per unit. It can be seen from fig. 1 that this Lindahl mechanism defined in net terms is formally equivalent to the Lindahl mechanism defined in gross terms:4 the four schedules labelled c --JJ y), MC,(y), d;(y) and d,(y) are defined relative to the origin 0,; relative to the origin 0, these same four schedules are --r&(y) [i.e. the average net cost of Y to individual 1, which is minus the announced net marginal willingness-to-pay (or evaluation) of individual 21, NMC,(y) (i.e. the marginal net cost of Y to 1, or d/dy{y[-r&(y)]}), G,(y) [i.e. l’s announced net marginal evaluation of Y] and nl(y) [i.e. l’s true net marginal evaluation of Y], respectively. These four schedules are drawn in fig. 2 which is obtained from fig. 1 merely by appropriately re-labelling the schedules and omitting the origins 0, and 0,. in generality in assuming that the average cost of Y in terms of X is a constant. Let C(y) denote the total cost in terms of X of y units of Y; if C”(y)LO, then the true net marginal evaluations must be defined as: ni(y)=di(y)-a/Z’(y),
i=l,2.
The announced net marginal evaluations must be defined analogously and the remainder of the analysis is unaffected. %ince the two mechanisms are equivalent, and since the basic shares in cost LYEand t(* do not appear in the tirst (gross) mechanism, it is clear that the allocation of public and private goods achieved by the second (net) mechanism is independent of these basic shares. In the mechanisms to be described in subsequent sections, however, these basic shares play an important part.
362
G. Fane and E. Sieper, Preference revelation and monopsony
NMC,(y)
Fig. 2
3. The demand revealing process Suppose now that the Lindahl mechanism of section 2 is amended by abolishing the side-payments between individuals 1 and 2; in place of these side-payments individual 1 has to pay the government the area below the curve, -F&(Y), up to the chosen level 9. Similarly, 2 must pay the government the area below --g,(y) up to 9. In addition, the two individuals continue to pay the government their assigned shares in total cost. By charging 1 the area below the curve, -AZ(y), he is in effect converted from a simple monopsonist into a perfectly discriminating monopsonist: - fi2( y) now represents his net marginal cost of x and his utility maximizing behavior involves equating this with his true net marginal evaluation of Y Since the government’s rule, eq. (1’) involves equating it to his announced net marginal evaluation of E: he has an incentive to announce his true net marginal evaluation. This is the essence of all the various mechanisms discussed in this paper for inducing truthful revelation of preferences. The proposal that individual 1 bc charged the area below the curve -c,(y) does not define a unique mechanism, since it involves an arbitrary constant of intergration; rather, it defines the class of Groves mechanisms. Clarke (1971) proposed a particular member of this class, which has the property of guaranteeing that the government receives at least enough revenue to finance the provision of the public good. This particular mechanism has been named the demand revealing process (DRP) by Tideman and Tullock (1976). Under the DRP the arbitrary constants of
G. Fane and E. Sieper, Preference revelation and monopsony
363
integration are defined as follows: let y”_i be the level of Y at which 2’s announced net marginal evaluation is zero, 5 i.e. &(j_ i) =O; similarly &2 is defined by A1(g_2) = 0. Individual 1 must pay the government the area under - Tz,(y) from y- 1 to y. This is referred to as the Clarke tax on 1; in fig. 3 this is the diagonally shaded triangular area. Individual 2 must pay the government the area below -rii(y) from ~7_~ to ~7. Given that 1 announces his true preferences it is clear that the absolute value of the Clarke tax on 2 is the vertically shaded triangular area in fig. 3. In this range --h,(y) is negative, but since j is less than j-2 the Clarke tax on 2 is positive. Since all individuals pay positive Clarke taxes as well as paying their assigned shares in the total cost of T: the government necessarily has a budget surplus.6
Fig. 3
Under the DRP, utility maximization only requires that each individual announce his true net marginal evaluation of Y at ~7. For example, in the case illustrated in fig. 3, individual l’s utility would be unaffected by the ‘In the general case, with I individuals, Y-i is the level of Y defined by: ~jtifj(j_i)=O. 6Fig. 3 shows that in the case in which iil(yT is positive and AZ(g) is negative, both individuals pay positive Clarke taxes. Obviously, we need only re-label the individuals to cover the case in which ril(i) is negative and &(yJ is positive. The only other possibility is that AI(P)=A2(yT=0. In this case the Clarke taxes are zero.
G. Fane and E. Sieper, Preference revelation and monopsony
364
rotation7 of his announced schedule about the intersection point of nr(y) and -A,(y). However, such a rotation would directly affect the Clarke tax paid by 2. In order to preserve the incentives to reveal preferences truthfully it is therefore essential that the Clarke taxes be wasted. The need for the assumption that there are no income effects in the demands for the public good also arises from the possibility of rotating the announced net marginal evaluation schedules. If income effects were present the schedules in fig. 3 could only be defined conditionally on the levels of consumption of the private good by each individual. This in itself would not present any intrinsic difficulty. However, individual 1, for example, could then benefit from rotating his announced schedule fii(y) about the intersection point between A,(y) and -fi,(y); fig. 3 shows that such a rotation would alter the Clarke tax paid by 2, and would therefore shi$ the -&(y) schedule if 2’s net marginal evaluation of Y depended on 2’s consumption of the private good. In the situation illustrated in fig. 3, individual 1 would gain from an increase in 2’s net marginal evaluation of Y at each level of output, i.e. from a downward shift in [ - iz,(y)]. On the other hand, 2 would gain from a reduction in l’s net marginal evaluation of Y at each level of output. If Y were a normal good for both individuals it would be in l’s self-interest to rotate -ii(y) in fig. 3 in a clockwise direction, thereby reducing 2’s Clarke tax, and in 2’s self-interest to rotate -fiZ(y) in a clockwise direction, thereby increasing l’s Clarke tax. We have not attempted to analyze the properties of the final equilibrium; what is clear is that the simple explanation of why truth-telling is a dominant strategy under the DRP breaks down if income effects are present in the demands for Y If it were possible to avoid wasting the Clarke taxes the net social gain from producing the public good would be measured by the horizontally shaded triangle in fig. 3. Given that the Clarke taxes must be wasted, the net social gain is this triangle less the combined Clarke tax triangles. Fig. 3 therefore illustrates the importance of the choice of the ai parameters: it is quite possible for these parameters to be chosen so that the area of the Clarke tax triangles exceeds the area of the horizontally shaded triangle, in which case social welfare would be higher if the public good were not produced at all than if it were produced under the DRP. The next two mechanisms which we analyze, the GLOM and the DRG, involve amending the DRP to attempt to avoid the need to waste the Clarke taxes.
4. The Groves-Ledyard The
GLOM
can
optimal mechanism
be represented
in the following
terms:
each
individual
‘This strategy [i.e. rotation of ri,(y)] is only available to 1 if he knows A,(y) before he announces 2,(y); otherwise he cannot be sure of the value of p ex ante, and his strictly dominant strategy would then be to announce his true preferences at every level of output.
G. Fane and E. Sieper, Preference revelation and monopsony
365
must announce his net marginal evaluation, ti,(y*), of Y at some particular output y* by sending a message m, (y*). to the government; the government implicitly calculates &(y*) from the formula:
rii( _V*)= ymi(y*) - YZ - ‘_)I*,
(2)
where y is an arbitrary positive constant chosen by the government, and I is the number of individuals. In effect, the government specifies the gradient, -?I-‘, of rii(y) and each individual’s message specifies the intercept, ymi(y*), which yields his announced net marginal evaluation at y*. The fact that the gradient of the yii(y) function is specified by the government prevents individuals from rotating their announced net marginal evaluation schedules, so as to affect the Clarke taxes paid by others. This means that the GLOM does not require income effects to be absent from the demand for the public good. An unspecified tatonnement process’ is used to find the output level, y”, at which: Tmi(F)=J;:
(3)
Despite its superficial unfamiliarity, this equation in fact represents the familiar efficiency condition (common to all the mechanisms which we discuss) for determining the optimal output of a public good, i.e. c MRS = MRT. The equivalence between eq. (3) and eq. (1’) can be seen by summing the right-hand side of eq. (2) over all individuals. Fig. 4 illustrates the equilibrium under the GLOM as viewed by individual 1; it is drawn on the supposition (to be justified below) that individual 1 does indeed send the message, m,(F), which reveals his true net marginal evaluation of Y at the output level jj, i.e. nl(y) and h,(y) intersect at ~7. In addition to his gross share in the total cost of the public good, al. C(q), individual 1 is also charged a Clarke tax calculated from (a) his announced net marginal evaluation schedule, G,(y), and (b) the sum of the announced net marginal evaluation schedules for all others, xii i i,(y). Henceforth we shall refer to this schedule as fi2(y) to highlight the analogy between the GLOM and the other mechanisms.’ This Clarke tax is calculated in the same way as for the 8Groves and Ledyard (1977, 1980) are mainly concerned with proofs of existence and optimality. A referee has drawn our attention to the apparent absence of a proof that there exists a tltonnement process which necessarily converges under the general assumptions made by Groves and Ledyard. However, at least in simple cases, it would not seem hard to construct such a proof; if, for example, there is only one public good and no income effects in the demand for it, then there is a unique optimal level of k: i.e. f, and if the auctioneer conducting the tltonnement process adjusts y* according to the rule dy*=I{~imi(y*)-y*}, where I>O, he will always be adjusting y* towards JY 9When the number of individuals, 1, exceeds two, it would also be necessary to replace t&(y) by I{= 2 &(y) in order to describe the mechanisms of sections 2 and 3 as they affect individual 1. The GLOM in fact requires that there be at least three individuals. This can be seen from the definition of af (see below).
G. Fune and E. Sieper, Preference revelation and monopsony
366
-i-l,(y).
slope = y(l-I)I-’
-n?(Y)
Fig. 4
DRP. It is the diagonally shaded area in fig. 4. By construction the height of the -n;(y) line above the horizontal axis at y” is uil(y”), which is y[m, -I-‘~71. Let ~1~ denote the average of the messages sent by everyone except 1. From (3) it therefore follows that:
Therefore
the height of l’s Clarke
tax triangle
in fig. 4 is:
Combining this infoimation with the fact that the gradient of the -A2(yJ is y(Z - 1)1-l, it follows that l’s Clarke tax is: sy(Z- l)Z-‘(ml -,L#.
line
The GLOM also differs from the DRP in that each individual is given a rebate of &~a:, where c$ is the variance of the messages sent by everyone except i:
d=[
&i(mh-Pi)i]/(z-4.
G. Fane and E. Sieper, Preference revelation and monopsony
367
The total tax paid by each individual is equal to: (a) his gross share in the total cost of the public good, plus (b) his Clarke tax, less (c) his rebate of &J:. It is straightforward to check that these three components correspond to the terms in the Groves-Ledyard formula,l’ and to prove that the sum of the rebates for all individuals is equal to the sum of the Clarke taxes for all individuals.” The government’s budget is therefore balanced. Groves and Ledyard assume ‘competitive behavior’, by which they mean that each individual treats the messages of all others as exogenously given. Under this assumption each individual regards his own rebate as a lump-sum payment, since it depends only on the messages of others. Individual 1 regards the -AZ(~) schedule as fixed; therefore its height measures the net marginal cost to him of Y and his utility maximizing strategy is to equate this net marginal cost to his true net marginal evaluation of Y ni(yJ. Given that the government’s rule equates it to izi(yJ, individual l’s utility maximizing strategy is to send the message which reveals his true net marginal evaluation. Therefore, given the competitive behavior assumption, the GLOM is efficient and produces a balanced budget.
5. Arrow’s demand revealing game Our exposition of the DRG is based on Arrow (1977). The setting for this mechanism is identical to that described in sections 2 and 3. The object of the DRG is to eliminate the waste of the Clarke taxes inherent in the DRP. This objective can be achieved if some extra information is available: each individual knows the utility function of the other once a particular vector of parameters is specified. Knowledge of the particular values of these parameters is assumed to be private, but knowledge of the joint density function from which they are drawn is assumed to be public. For simplicity, we assume that individual 2’s net marginal evaluation function of Y depends on a single shift parameter, p2: 122 =
n2b
P2).
Individual 1 knows the functional form of nz( .), but not the particular value of p2; however, he does know that p2 was drawn randomly from a population with a density function f2(p2), which is objectively known to everyone. There are known to be no income effects in the demand for the ‘“Eq. (4.3) (c), p. 796 in Groves terms of our notation this tax is: z,v”+(y/2) “See
Groves
{(I-
and Ledyard
1)1-‘(m,-~;,‘-cT’}
and Ledyard
(1977, p. 798).
(1977), gives the tax payable
by individual
i. In
368
G. Fane and E. Sieper, Preference
public good. Exactly symmetrical of l’s net marginal evaluation of value of his own parameter. Each individual announces government chooses the output, net marginal evaluations is zero:
revelation
and monopsony
assumptions are made about 2’s knowledge Y; and of course each individual knows the the value his own parameter, and the Y; of Y at which the sum of the announced
where /jl and bZ denote the values announced by 1 and 2, respectively. The problem is to design side-payments to induce each individual to announce the true value of his parameter. Suppose that once b1 and b2 had been announced we calculated the Clarke taxes from the resulting functions, ni(y,br) and nz(y,fi2), and then wasted the Clarke taxes. This would be the DRP under uncertainty; under the DRP the incentive to each individual to reveal his true preferences is independent of the particular preferences of the other. Therefore uncertainty about what the other will reveal would not influence the true announcement which each would make. Now suppose that, in order to avoid the waste of the Clarke taxes inherent in the DRP, we required each individual to pay his own Clarke tax to the other. The defect with this suggestion is that by varying his own announcement each individual has the power to affect the other’s Clarke tax; provided that the Clarke taxes are wasted (as in section 3) this does not matter, but in the procedure suggested above it would clearly provide each with an incentive to make a false announcement. If there was some way of making the Clarke tax paid by each independent of the other individual’s announcement this defect would not arise. For individual 1 we need to calculate a Clarke tax which depends on fil but not on b2. The DRG achieves this by calculating l’s Clarke tax from the functions n,(y, fil) and n,(y, pJ for every value of pZ and then integrating over pZ, weighting each Clarke tax outcome for 1 with the objectively known density functionf2(P2). Individual 2 is treated in an exactly symmetrical way. Clearly, the Clarke tax paid by each is now independent of the other’s announcement and, instead of being wasted, the Clarke taxes of each can be rebated to the other without distorting incentives. Provided that 1 is risk-neutral and maximizes expected utility his incentive to reveal his true parameter is not changed by replacing his Clarke tax calculated from (fir, jj2) by the Clarke tax calculated from (pi, pJ integrated over f&J. This can be proved as follows: suppose that 1 and 2 are both making truthful announcements, but that 1 now considers making a marginal change in his 1 would raise his expected true own announcement. By raising fii, individual net evaluation of Y at the rate:
G. Fane and E. Sieper, Preference revelation and monopsony
Individual
l’s expected
Therefore
by raising
Clarke
369
tax is given by:
j?r, 1 would
raise his expected
Given that 2 is certain to tell the relationship must be satisfied:
truth,
Clarke
1 knows
tax at the rate:
that
the
following
nl(~(Pl,P2),~1)+122(Y”(Pl,P2),P2)=0. To maximize utility 1 must (7) into (6) this gives:
equate
expressions
(5) and (6); substituting
(7) from
This first-order condition is obviously satisfied if 1 sets fii =pr, i.e. truthtelling is the utility maximizing strategy for 1, given that 2 is certain to tell the truth. Individual 2’s side-payment is calculated in analogous manner and therefore his best strategy is to tell the truth, if he is certain that 1 will be truthful. This establishes that truth-telling is an equilibrium in the DRG.
6. The theoretical
limitations
of the DRP, the GLOM and the DRG
A large and growing literature, which we do not attempt to survey, is concerned with the sense in which any mechanism must inherently suffer from some limitations. This section merely attempts to note the main limitations of the DRP, the GLOM and the DRG. All these mechanisms ignore the possible formation of coalitions. However, coalitions are of course vulnerable to free-rider problems of their own. In addition, the DRP suffers from three defects: first, the Clarke taxes must be wasted if incentives are not to be distorted; furthermore, this waste may be so large that social welfare is lower with the public good produced and financed by the DRP than it would have been if the public good had not been produced at all. Second, for the reasons given in section 3, there must be no income effects in the demand for the public good. Third, when there are many individuals, the variations in any one individual’s utility which
370
G. Fane and E. Sieper, Preference revelation and monopsony
result from fine calculations of the marginal costs and benefits of different announcements is likely to be negligible when compared to the cost of making these calculations. This last defect applies also to the GLOM, and since this mechanism requires an iterative procedure to be followed in order to find the final equilibrium the defect is likely to be particularly important in this case. A second defect of the GLOM is that, at least in the absence of arbitrarily large numbers of individuals, it will be possible for each to affect the messages sent by the others. This is clear from fig. 4: by raising his own message 1 will raise the level of j and xi= 2 mi will rise (fall) if the slope of c I=, - ni(y) is less (greater) than y(I- 1)1- ‘. While y could, perhaps, be chosen to make these slopes equal, so that individual 1 could not influence the sum of the messages sent by all others, it would then generally be the case that individuals 2,. . ., I could influence the messages sent by all others. The ability of any one individual to alter the messages sent by others creates incentives to distort preferences under the GLOM in the same way that income effects create incentives to distort preferences under the DRP. Both situations provide individual 1 with the opportunity to shift the -&(y) schedule. When all individuals behave ‘competitively’ (i.e. treat the messages of others as exogenous) then any one individual, for example the first, could calculate his optimal deviation from competitive behavior if he knew the true preferences of all the others, but it is the essence of the free-rider problem that this information is not available. However, if all others behave competitively the first can estimate the relevant information by experimentally changing his own message during the tgtonnement process, and observing how the mean and variance of the messages of all others vary with the output of the public good. The situation in which all individuals behave ‘competitively’ except the first, while he maximizes his own utility, taking account of the predictable variations in the behavior of others, is closely analogous to the situation in oligopoly in which the policy of all firms except the first is to follow a Cournot-style reaction function, while the first firm maximizes profits subject to the constraints that the other firms remain on their reaction functions: the first firm is a Stackelberg leader and the others are Stackelberg followers. I2 Because of its asymmetry the Stackelberg leader-followers model is not a satisfactory solution to the oligopoly problem, although it provides a legitimate criticism of the assumption that oligopolists follow reaction functions. Similarly, we are not claiming that under the GLOM one person would lead and the rest follow, but only that the assumption of competitive behavior implies that at least “See
Fellner (1949, ch. III) for an exposition
of Stackelberg’s
theory
of oligopoly.
G. Fane and E. Sieper, Prej&-ence recelation and monopsony
371
some individuals are behaving like ‘followers’ when they could have greater utility as ‘leaders’.r3 A case can be made that the Cournot solution does solve the problem of oligopoly, not because firms follow reaction functions, but because Cournot’s solution is a Nash equilibrium for the non-cooperative oligopoly game.r4 Similarly, GL assert that the equilibrium under their assumption of competitive behavior is a Nash equilibrium.r5 However, the analogy is misleading: if the oligopoly game is played only once, then, in the Nash equilibrium, firms immediately pick the intersection point of the Cournot reaction functions, but these strategies are not picked for Cournot’s reason, since reacting is not possible in a one-move game; for the same reason leading and following are also impossible, and Stackelberg’s criticism of Cournot is therefore irrelevant. However, the crucial difference between the oligopoly problem and the free-rider problem is that in the former every player knows all the possible pay-offs to himself and to others, while in the latter the pay-offs to others are not known. Therefore, in the oligopoly models for which the intersection of the Cournot reaction functions is also the equilibrium of the Nash non-cooperative game there is no tdtonnement process, but in the free-rider problem under the GLOM such a process is essential, since competitive behavior, in GL’s sense, is not based on game theory but involves behaving like a firm following a reaction function. Like the DRP, the DRG breaks down if there are income effects in the demand for the public good. A second limitation of the DRG is the requirement that the probability density functions of the parameters which determine preferences must be objectively known. Finally, it is important to remember that the mere existence of theoretical limitations to the efficiency of these three mechanisms is not necessarily inconsistent with the absence of any serious practical defects; however, the apparent absence of any examples of the real-world use of any of these mechanisms suggests that they do suffer from serious practical defects. “Groves and Ledyard are of course aware model [Groves and Ledyard (1977, p. 807)]. ‘%ee Friedman (1977, pp. 168-171). “See Groves and Ledyard (1977, p. 807).
that competitive
behavior
is not optimal
in their
References Arrow, K.J., 1977, The property rights doctrine and demand revelation under incomplete information, Institute for Mathematical Studies in the Social Sciences, Stanford University, Technical Report No. 243. Clarke, E.H., 1971, Multipart pricing of public goods, Public Choice 11, 17-33. Fellner, W.J., 1949, Competition among the few (A.A. Knopf, New York). Friedman, J.W., 1977, Oligopoly and the theory of games (North-Holland, Amsterdam). Groves, T., 1973, Incentivies in teams, Econometrica 41. 617-633. Groves, T. and J.O. Ledyard, 1977, Optimal allocation of public goods: A solution to the freerider problem, Econometrica 45, 7833810.
312
G. Fane and E. Sieper, Prrjerence revelation and monopsony
Groves, T. and J.O. Ledyard, 1980, The existence of efficient and incentive compatible equilibria with public goods, Econometrica 48, 1487-1506. Groves, T. and M. Loeb, 1975, Incentives and public inputs, Journal of Public Economics 4, 21 l-226. Johnson, H.G., 1977, A note on the theory of public goods and ‘bluffing’, Journal of Public Economics 7, 3833386. Tideman, T.N. and G. Tullock, 1976, A new and superior process for making social choices, Journal of Political Economy 84, 1145-l 159. Vickrey, W., 1961, Counterspeculation, auctions and competitive sealed tenders, Journal of Finance 16. 8-37.
of Public
Economics
20 (1983) 357-372.
PREFERENCE
North-Holland
REVELATION
Publishing
AND
Company
MONOPSONY
G. FANE University
of Victoria and Australian
National
University,
Canberra, A.C.?: 2600, Australia
E. SIEPER Australian
Received
National
February
University,
Canberra, A.C.7: 2600, Australia
1981, revised version
received
November
1981
(1) Johnson demonstrated the formal equivalence between the free-rider problem and monopsony. (2) Vickrey’s method for eliminating the inefliciencies resulting from monopsony (monopoly) works by converting simple monopsonists (monopolists) into perfectly discriminating monopsonists (monopolists). (3) We show that three recently proposed solutions to the free-rider problem have the same rationale, i.e. they use ‘Clarke taxes’ to convert Johnson’s simple monopsonists into perfectly discriminating monopsonists. The three mechanisms are those of: (i) Clarke and Groves; (ii) Groves and Ledyard; and (iii) Arrow. The relationships between these mechanisms, and between each of them and monopsony, are analyzed diagrammatically.
1. Introduction The purpose of this paper is to synthesize some recent contributions to the literature on preference revelation and the free-rider problem. Our basic point is that all the preference revealing mechanisms which we discuss are best understood in terms of the theory of monopsony, and all can be analyzed using the same simple geometric approach. Johnson (1977) noted the formal equivalence between the free-rider problem and monopsony: when a consumer-taxpayer is invited to state his willingness-to-pay for a public good and charged accordingly he is in the position of a monopsonist; his self-interest involves ‘free-riding’, i.e. the understatement of his true willingness-to-pay for the public good so as to reduce the average price which he pays. This leads to an inefficiently low level of output of the public good for just the same reasons that simple monopsony leads to inefficiently low output in markets for private goods. In the next section of this paper we explain this idea of Johnson’s in more detail, and in section 3 we show that the demand revealing process [i.e. the mechanism due to Clarke (1971), Groves (1973) and Groves and Loeb (1975)], henceforth the DRP, succeeds in inducing truthful preference revelation because it converts the simple monopsonists of Johnson’s problem 0047-2727/83/$03.00
0
Elsevier Science Publishers
B.V. (North-Holland)
358
G. Fane and E. Sieper, Preference revelation and monopsony
into perfectly discriminating monopsonists. It is of course well known that perfectly discriminating monopsony avoids the inefficiencies which result from simple monopsony. The formal equivalence between the DRP and the method proposed by Vickrey (1961) for overcoming the inefficiencies caused by monopsonies and monopolies is well known.’ In essence Vickrey’s method is to convert simple monopolists into perfectly discriminating monopolists by paying to each the area under his demand curve, and to convert simple monopsonists into perfectly discriminating monopsonists by charging each the area under his supply curve. The role of the so-called ‘Clarke taxes’ in the DRP is to ensure that Johnson’s representative monopsonist is charged the area under his supply curve, rather than the revenue rectangle defined by the chosen point on this curve. Given Johnson’s demonstration of the formal equivalence between the free-rider problem and monopsony it is not surprising that Vickrey’s method should also be the basis for the mechanisms for overcoming the free-rider problem. The existence of a relationship between the DRP and discriminating monopoly was noted by Groves (1973). In his model the resource manager of a conglomerate-type organization in which there are production externalities between the constituent sub-units attempts to set prices for each activity and for each sub-unit so as to maximize profits. After defining the ‘prices’ which yield optimal incentives, Groves notes that: While it is perhaps suggestive to represent the own profit incentive structure in terms of prices charged for resources allocated, it should be observed that these prices are in general not only different for the different enterprises, but also that the rules (functions) by which they are calculated are different for each enterprise. This non-anonymity property is unconventional, but has some parallel to the situation of a discriminating monopolist [Groves (1973, p. 629) emphasis added]. In sections 4 and 5 we show that the Groves-Ledyard optimal mechanism [Groves and Ledyard (1977, 1980)] henceforth the GLOM, and the demand revealing game, DRG [Arrow (1977)] also implicitly use Clarke taxes to convert simple monopsonists into perfectly discriminating monopsonists. Interpreting the GLOM and the DRG in terms of Clarke taxes allows us to exposit these mechanisms in terms of our basic geometric analysis of monopsony, and also helps to clarify the apparently disparate relationships between these mechanisms, the DRP and the contributions of Johnson and Vickrey. Finally, our analysis reveals why a necessary assumption for truthful revelation of preferences under the DRP is that there are no income effects in ‘See Tideman and Tullock (1976, p. 1146).
G. Fane and E. Sieper, Preference revelation and monopsony
the demand for the public good, and why this assumption for truthful revelation of preferences under the GLOM.
359
is not necessary
2. The free-rider problem as an instance of simple monopsony We begin this section with Johnson’s demonstration of the formal equivalence between the free-rider problem and simple monopsony. For simplicity, consider an economy with one private good, X, one public good, I: and just two consumer-taxpayers. Their true marginal evaluations of the public good are d,(y) and d2(y). Throughout sections 2, 3 and 5 we assume that each individual’s marginal evaluation of Y is independent of his level of consumption of the private good.2 The marginal cost of the public good is a constant, c. The government determines the output of the public good by asking each individual to announce his marginal evaluation and selecting the output level, Y; at which the sum of the announced marginal evaluations is equal to the marginal cost of the public good:
W)+&(y”)=c,
(1)
where &y”) denotes individual i’s announced marginal evaluation. The output of the public good is financed by requiring each individual to pay an average price for all units of Y equal to his announced marginal evaluation of the last unit produced. This method of providing public goods is usually referred to as the Lindahl mechanism. It is clear that the revenue raised is just sufficient to finance the output of the public good, leaving the government with a balanced budget. If each individual announces his true marginal evaluation schedule then the mechanism also achieves a Pareto efficient allocation, since eq. (1) then becomes the familiar efficiency condition c MRS= MRT. However, as Johnson pointed out, each individual has an incentive to announce a false marginal evaluation schedule because each is in the position of a monopsonist. This point is clear from fig. 1, in which amounts of the public good are measured on the abscissa and per unit evaluations and costs of the public good are measured on the ordinate. The vertical distance between 0, and 0, is c. With 0, as origin the curve labelled d,(y) represents individual l’s true marginal evaluation of Y; the curve labelled c -J2(y) can be interpreted in two ways: with 0, as origin and reading downwards it represents individual 2’s announced marginal evaluation of x with 0, as origin and reading upwards it represents the average price which individual 1 must pay per unit of Y as a function of the amount of Y.which individual 1 obtains, given the government’s rules. If individual 1 were to announce his true marginal evaluation schedule, d,(y), the government would select output ‘It is assumed that i’s utility function can be written in the form: ui( y,x,) = ui(y)+xi, true marginal evaluation of I: in terms of X, is given by: di( y) = vx y).
so that i’s
360
G. Fane and E. Sieper, Preference revelation and monopsony
MC,(Y)
c-
4
1,”
?
Fig. 1
level y*. However, given the schedule J2(y), individual 1 has an incentive free ride: the marginal cost of Y to individual 1, MC,(y), is given by:
to
This schedule, which is the marginal schedule derived from l’s average price schedule, is drawn in fig. 1. By announcing a false schedule, such as d;(y), individual 1 maximizes utility, since this schedule achieves the output level, p, at which l’s true marginal evaluation of Y is equal to the private marginal cost of Y to 1. By acting in this way 1 is exercising his monopsony power. For the purposes of comparing the DRP, the GLOM and the DRG with each other, and for showing how each of these mechanisms attempts to overcome the problem of monopsony in the Lindahl mechanism, it is helpful to redefine the Lindahl mechanism in net terms rather than gross terms: whereas di(y) and &(y) denote individual i’s true and announced marginal evaluation schedules for the public good, we now define ni( y) and yii(y) as his true and announced net marginal evaluations of the public good given that the government requires individual i to pay a share xi in the cost of providing the public good. These shares are determined so that xiai= 1. The true net marginal evaluations are:3 3An additional
advantage
of working
in net terms is that it clearly
reveals that there is no loss
G. Fane and E. Sieper, Preference revelation and monopson)
ni(Y)=di(Y)-aic, The announced
i=
net marginal
42.
evaluations i=
Ai(y)=~i(y)-_cl,C,
are:
1,2.
The government selects the output marginal evaluations sum to zero: &(Jq+ti,(y~=O.
361
level, 9, at which
the announced
net
(1’)
Individual 1 is required to make a side-payment of A,(j) to individual 2 for every unit of Y produced. Therefore the total payment by 1, per unit of Y is:
Similarly, individual 2 can be thought of as making a side-payment of &(y”), per unit of Y produced, to individual 1. Since &(y’)= -izi(y?, these two methods of defining the side-payment are equivalent definitions of a single transfer. The gross cost of Y to individual 2 is still a,(y7) per unit. It can be seen from fig. 1 that this Lindahl mechanism defined in net terms is formally equivalent to the Lindahl mechanism defined in gross terms:4 the four schedules labelled c --JJ y), MC,(y), d;(y) and d,(y) are defined relative to the origin 0,; relative to the origin 0, these same four schedules are --r&(y) [i.e. the average net cost of Y to individual 1, which is minus the announced net marginal willingness-to-pay (or evaluation) of individual 21, NMC,(y) (i.e. the marginal net cost of Y to 1, or d/dy{y[-r&(y)]}), G,(y) [i.e. l’s announced net marginal evaluation of Y] and nl(y) [i.e. l’s true net marginal evaluation of Y], respectively. These four schedules are drawn in fig. 2 which is obtained from fig. 1 merely by appropriately re-labelling the schedules and omitting the origins 0, and 0,. in generality in assuming that the average cost of Y in terms of X is a constant. Let C(y) denote the total cost in terms of X of y units of Y; if C”(y)LO, then the true net marginal evaluations must be defined as: ni(y)=di(y)-a/Z’(y),
i=l,2.
The announced net marginal evaluations must be defined analogously and the remainder of the analysis is unaffected. %ince the two mechanisms are equivalent, and since the basic shares in cost LYEand t(* do not appear in the tirst (gross) mechanism, it is clear that the allocation of public and private goods achieved by the second (net) mechanism is independent of these basic shares. In the mechanisms to be described in subsequent sections, however, these basic shares play an important part.
362
G. Fane and E. Sieper, Preference revelation and monopsony
NMC,(y)
Fig. 2
3. The demand revealing process Suppose now that the Lindahl mechanism of section 2 is amended by abolishing the side-payments between individuals 1 and 2; in place of these side-payments individual 1 has to pay the government the area below the curve, -F&(Y), up to the chosen level 9. Similarly, 2 must pay the government the area below --g,(y) up to 9. In addition, the two individuals continue to pay the government their assigned shares in total cost. By charging 1 the area below the curve, -AZ(y), he is in effect converted from a simple monopsonist into a perfectly discriminating monopsonist: - fi2( y) now represents his net marginal cost of x and his utility maximizing behavior involves equating this with his true net marginal evaluation of Y Since the government’s rule, eq. (1’) involves equating it to his announced net marginal evaluation of E: he has an incentive to announce his true net marginal evaluation. This is the essence of all the various mechanisms discussed in this paper for inducing truthful revelation of preferences. The proposal that individual 1 bc charged the area below the curve -c,(y) does not define a unique mechanism, since it involves an arbitrary constant of intergration; rather, it defines the class of Groves mechanisms. Clarke (1971) proposed a particular member of this class, which has the property of guaranteeing that the government receives at least enough revenue to finance the provision of the public good. This particular mechanism has been named the demand revealing process (DRP) by Tideman and Tullock (1976). Under the DRP the arbitrary constants of
G. Fane and E. Sieper, Preference revelation and monopsony
363
integration are defined as follows: let y”_i be the level of Y at which 2’s announced net marginal evaluation is zero, 5 i.e. &(j_ i) =O; similarly &2 is defined by A1(g_2) = 0. Individual 1 must pay the government the area under - Tz,(y) from y- 1 to y. This is referred to as the Clarke tax on 1; in fig. 3 this is the diagonally shaded triangular area. Individual 2 must pay the government the area below -rii(y) from ~7_~ to ~7. Given that 1 announces his true preferences it is clear that the absolute value of the Clarke tax on 2 is the vertically shaded triangular area in fig. 3. In this range --h,(y) is negative, but since j is less than j-2 the Clarke tax on 2 is positive. Since all individuals pay positive Clarke taxes as well as paying their assigned shares in the total cost of T: the government necessarily has a budget surplus.6
Fig. 3
Under the DRP, utility maximization only requires that each individual announce his true net marginal evaluation of Y at ~7. For example, in the case illustrated in fig. 3, individual l’s utility would be unaffected by the ‘In the general case, with I individuals, Y-i is the level of Y defined by: ~jtifj(j_i)=O. 6Fig. 3 shows that in the case in which iil(yT is positive and AZ(g) is negative, both individuals pay positive Clarke taxes. Obviously, we need only re-label the individuals to cover the case in which ril(i) is negative and &(yJ is positive. The only other possibility is that AI(P)=A2(yT=0. In this case the Clarke taxes are zero.
G. Fane and E. Sieper, Preference revelation and monopsony
364
rotation7 of his announced schedule about the intersection point of nr(y) and -A,(y). However, such a rotation would directly affect the Clarke tax paid by 2. In order to preserve the incentives to reveal preferences truthfully it is therefore essential that the Clarke taxes be wasted. The need for the assumption that there are no income effects in the demands for the public good also arises from the possibility of rotating the announced net marginal evaluation schedules. If income effects were present the schedules in fig. 3 could only be defined conditionally on the levels of consumption of the private good by each individual. This in itself would not present any intrinsic difficulty. However, individual 1, for example, could then benefit from rotating his announced schedule fii(y) about the intersection point between A,(y) and -fi,(y); fig. 3 shows that such a rotation would alter the Clarke tax paid by 2, and would therefore shi$ the -&(y) schedule if 2’s net marginal evaluation of Y depended on 2’s consumption of the private good. In the situation illustrated in fig. 3, individual 1 would gain from an increase in 2’s net marginal evaluation of Y at each level of output, i.e. from a downward shift in [ - iz,(y)]. On the other hand, 2 would gain from a reduction in l’s net marginal evaluation of Y at each level of output. If Y were a normal good for both individuals it would be in l’s self-interest to rotate -ii(y) in fig. 3 in a clockwise direction, thereby reducing 2’s Clarke tax, and in 2’s self-interest to rotate -fiZ(y) in a clockwise direction, thereby increasing l’s Clarke tax. We have not attempted to analyze the properties of the final equilibrium; what is clear is that the simple explanation of why truth-telling is a dominant strategy under the DRP breaks down if income effects are present in the demands for Y If it were possible to avoid wasting the Clarke taxes the net social gain from producing the public good would be measured by the horizontally shaded triangle in fig. 3. Given that the Clarke taxes must be wasted, the net social gain is this triangle less the combined Clarke tax triangles. Fig. 3 therefore illustrates the importance of the choice of the ai parameters: it is quite possible for these parameters to be chosen so that the area of the Clarke tax triangles exceeds the area of the horizontally shaded triangle, in which case social welfare would be higher if the public good were not produced at all than if it were produced under the DRP. The next two mechanisms which we analyze, the GLOM and the DRG, involve amending the DRP to attempt to avoid the need to waste the Clarke taxes.
4. The Groves-Ledyard The
GLOM
can
optimal mechanism
be represented
in the following
terms:
each
individual
‘This strategy [i.e. rotation of ri,(y)] is only available to 1 if he knows A,(y) before he announces 2,(y); otherwise he cannot be sure of the value of p ex ante, and his strictly dominant strategy would then be to announce his true preferences at every level of output.
G. Fane and E. Sieper, Preference revelation and monopsony
365
must announce his net marginal evaluation, ti,(y*), of Y at some particular output y* by sending a message m, (y*). to the government; the government implicitly calculates &(y*) from the formula:
rii( _V*)= ymi(y*) - YZ - ‘_)I*,
(2)
where y is an arbitrary positive constant chosen by the government, and I is the number of individuals. In effect, the government specifies the gradient, -?I-‘, of rii(y) and each individual’s message specifies the intercept, ymi(y*), which yields his announced net marginal evaluation at y*. The fact that the gradient of the yii(y) function is specified by the government prevents individuals from rotating their announced net marginal evaluation schedules, so as to affect the Clarke taxes paid by others. This means that the GLOM does not require income effects to be absent from the demand for the public good. An unspecified tatonnement process’ is used to find the output level, y”, at which: Tmi(F)=J;:
(3)
Despite its superficial unfamiliarity, this equation in fact represents the familiar efficiency condition (common to all the mechanisms which we discuss) for determining the optimal output of a public good, i.e. c MRS = MRT. The equivalence between eq. (3) and eq. (1’) can be seen by summing the right-hand side of eq. (2) over all individuals. Fig. 4 illustrates the equilibrium under the GLOM as viewed by individual 1; it is drawn on the supposition (to be justified below) that individual 1 does indeed send the message, m,(F), which reveals his true net marginal evaluation of Y at the output level jj, i.e. nl(y) and h,(y) intersect at ~7. In addition to his gross share in the total cost of the public good, al. C(q), individual 1 is also charged a Clarke tax calculated from (a) his announced net marginal evaluation schedule, G,(y), and (b) the sum of the announced net marginal evaluation schedules for all others, xii i i,(y). Henceforth we shall refer to this schedule as fi2(y) to highlight the analogy between the GLOM and the other mechanisms.’ This Clarke tax is calculated in the same way as for the 8Groves and Ledyard (1977, 1980) are mainly concerned with proofs of existence and optimality. A referee has drawn our attention to the apparent absence of a proof that there exists a tltonnement process which necessarily converges under the general assumptions made by Groves and Ledyard. However, at least in simple cases, it would not seem hard to construct such a proof; if, for example, there is only one public good and no income effects in the demand for it, then there is a unique optimal level of k: i.e. f, and if the auctioneer conducting the tltonnement process adjusts y* according to the rule dy*=I{~imi(y*)-y*}, where I>O, he will always be adjusting y* towards JY 9When the number of individuals, 1, exceeds two, it would also be necessary to replace t&(y) by I{= 2 &(y) in order to describe the mechanisms of sections 2 and 3 as they affect individual 1. The GLOM in fact requires that there be at least three individuals. This can be seen from the definition of af (see below).
G. Fune and E. Sieper, Preference revelation and monopsony
366
-i-l,(y).
slope = y(l-I)I-’
-n?(Y)
Fig. 4
DRP. It is the diagonally shaded area in fig. 4. By construction the height of the -n;(y) line above the horizontal axis at y” is uil(y”), which is y[m, -I-‘~71. Let ~1~ denote the average of the messages sent by everyone except 1. From (3) it therefore follows that:
Therefore
the height of l’s Clarke
tax triangle
in fig. 4 is:
Combining this infoimation with the fact that the gradient of the -A2(yJ is y(Z - 1)1-l, it follows that l’s Clarke tax is: sy(Z- l)Z-‘(ml -,L#.
line
The GLOM also differs from the DRP in that each individual is given a rebate of &~a:, where c$ is the variance of the messages sent by everyone except i:
d=[
&i(mh-Pi)i]/(z-4.
G. Fane and E. Sieper, Preference revelation and monopsony
367
The total tax paid by each individual is equal to: (a) his gross share in the total cost of the public good, plus (b) his Clarke tax, less (c) his rebate of &J:. It is straightforward to check that these three components correspond to the terms in the Groves-Ledyard formula,l’ and to prove that the sum of the rebates for all individuals is equal to the sum of the Clarke taxes for all individuals.” The government’s budget is therefore balanced. Groves and Ledyard assume ‘competitive behavior’, by which they mean that each individual treats the messages of all others as exogenously given. Under this assumption each individual regards his own rebate as a lump-sum payment, since it depends only on the messages of others. Individual 1 regards the -AZ(~) schedule as fixed; therefore its height measures the net marginal cost to him of Y and his utility maximizing strategy is to equate this net marginal cost to his true net marginal evaluation of Y ni(yJ. Given that the government’s rule equates it to izi(yJ, individual l’s utility maximizing strategy is to send the message which reveals his true net marginal evaluation. Therefore, given the competitive behavior assumption, the GLOM is efficient and produces a balanced budget.
5. Arrow’s demand revealing game Our exposition of the DRG is based on Arrow (1977). The setting for this mechanism is identical to that described in sections 2 and 3. The object of the DRG is to eliminate the waste of the Clarke taxes inherent in the DRP. This objective can be achieved if some extra information is available: each individual knows the utility function of the other once a particular vector of parameters is specified. Knowledge of the particular values of these parameters is assumed to be private, but knowledge of the joint density function from which they are drawn is assumed to be public. For simplicity, we assume that individual 2’s net marginal evaluation function of Y depends on a single shift parameter, p2: 122 =
n2b
P2).
Individual 1 knows the functional form of nz( .), but not the particular value of p2; however, he does know that p2 was drawn randomly from a population with a density function f2(p2), which is objectively known to everyone. There are known to be no income effects in the demand for the ‘“Eq. (4.3) (c), p. 796 in Groves terms of our notation this tax is: z,v”+(y/2) “See
Groves
{(I-
and Ledyard
1)1-‘(m,-~;,‘-cT’}
and Ledyard
(1977, p. 798).
(1977), gives the tax payable
by individual
i. In
368
G. Fane and E. Sieper, Preference
public good. Exactly symmetrical of l’s net marginal evaluation of value of his own parameter. Each individual announces government chooses the output, net marginal evaluations is zero:
revelation
and monopsony
assumptions are made about 2’s knowledge Y; and of course each individual knows the the value his own parameter, and the Y; of Y at which the sum of the announced
where /jl and bZ denote the values announced by 1 and 2, respectively. The problem is to design side-payments to induce each individual to announce the true value of his parameter. Suppose that once b1 and b2 had been announced we calculated the Clarke taxes from the resulting functions, ni(y,br) and nz(y,fi2), and then wasted the Clarke taxes. This would be the DRP under uncertainty; under the DRP the incentive to each individual to reveal his true preferences is independent of the particular preferences of the other. Therefore uncertainty about what the other will reveal would not influence the true announcement which each would make. Now suppose that, in order to avoid the waste of the Clarke taxes inherent in the DRP, we required each individual to pay his own Clarke tax to the other. The defect with this suggestion is that by varying his own announcement each individual has the power to affect the other’s Clarke tax; provided that the Clarke taxes are wasted (as in section 3) this does not matter, but in the procedure suggested above it would clearly provide each with an incentive to make a false announcement. If there was some way of making the Clarke tax paid by each independent of the other individual’s announcement this defect would not arise. For individual 1 we need to calculate a Clarke tax which depends on fil but not on b2. The DRG achieves this by calculating l’s Clarke tax from the functions n,(y, fil) and n,(y, pJ for every value of pZ and then integrating over pZ, weighting each Clarke tax outcome for 1 with the objectively known density functionf2(P2). Individual 2 is treated in an exactly symmetrical way. Clearly, the Clarke tax paid by each is now independent of the other’s announcement and, instead of being wasted, the Clarke taxes of each can be rebated to the other without distorting incentives. Provided that 1 is risk-neutral and maximizes expected utility his incentive to reveal his true parameter is not changed by replacing his Clarke tax calculated from (fir, jj2) by the Clarke tax calculated from (pi, pJ integrated over f&J. This can be proved as follows: suppose that 1 and 2 are both making truthful announcements, but that 1 now considers making a marginal change in his 1 would raise his expected true own announcement. By raising fii, individual net evaluation of Y at the rate:
G. Fane and E. Sieper, Preference revelation and monopsony
Individual
l’s expected
Therefore
by raising
Clarke
369
tax is given by:
j?r, 1 would
raise his expected
Given that 2 is certain to tell the relationship must be satisfied:
truth,
Clarke
1 knows
tax at the rate:
that
the
following
nl(~(Pl,P2),~1)+122(Y”(Pl,P2),P2)=0. To maximize utility 1 must (7) into (6) this gives:
equate
expressions
(5) and (6); substituting
(7) from
This first-order condition is obviously satisfied if 1 sets fii =pr, i.e. truthtelling is the utility maximizing strategy for 1, given that 2 is certain to tell the truth. Individual 2’s side-payment is calculated in analogous manner and therefore his best strategy is to tell the truth, if he is certain that 1 will be truthful. This establishes that truth-telling is an equilibrium in the DRG.
6. The theoretical
limitations
of the DRP, the GLOM and the DRG
A large and growing literature, which we do not attempt to survey, is concerned with the sense in which any mechanism must inherently suffer from some limitations. This section merely attempts to note the main limitations of the DRP, the GLOM and the DRG. All these mechanisms ignore the possible formation of coalitions. However, coalitions are of course vulnerable to free-rider problems of their own. In addition, the DRP suffers from three defects: first, the Clarke taxes must be wasted if incentives are not to be distorted; furthermore, this waste may be so large that social welfare is lower with the public good produced and financed by the DRP than it would have been if the public good had not been produced at all. Second, for the reasons given in section 3, there must be no income effects in the demand for the public good. Third, when there are many individuals, the variations in any one individual’s utility which
370
G. Fane and E. Sieper, Preference revelation and monopsony
result from fine calculations of the marginal costs and benefits of different announcements is likely to be negligible when compared to the cost of making these calculations. This last defect applies also to the GLOM, and since this mechanism requires an iterative procedure to be followed in order to find the final equilibrium the defect is likely to be particularly important in this case. A second defect of the GLOM is that, at least in the absence of arbitrarily large numbers of individuals, it will be possible for each to affect the messages sent by the others. This is clear from fig. 4: by raising his own message 1 will raise the level of j and xi= 2 mi will rise (fall) if the slope of c I=, - ni(y) is less (greater) than y(I- 1)1- ‘. While y could, perhaps, be chosen to make these slopes equal, so that individual 1 could not influence the sum of the messages sent by all others, it would then generally be the case that individuals 2,. . ., I could influence the messages sent by all others. The ability of any one individual to alter the messages sent by others creates incentives to distort preferences under the GLOM in the same way that income effects create incentives to distort preferences under the DRP. Both situations provide individual 1 with the opportunity to shift the -&(y) schedule. When all individuals behave ‘competitively’ (i.e. treat the messages of others as exogenous) then any one individual, for example the first, could calculate his optimal deviation from competitive behavior if he knew the true preferences of all the others, but it is the essence of the free-rider problem that this information is not available. However, if all others behave competitively the first can estimate the relevant information by experimentally changing his own message during the tgtonnement process, and observing how the mean and variance of the messages of all others vary with the output of the public good. The situation in which all individuals behave ‘competitively’ except the first, while he maximizes his own utility, taking account of the predictable variations in the behavior of others, is closely analogous to the situation in oligopoly in which the policy of all firms except the first is to follow a Cournot-style reaction function, while the first firm maximizes profits subject to the constraints that the other firms remain on their reaction functions: the first firm is a Stackelberg leader and the others are Stackelberg followers. I2 Because of its asymmetry the Stackelberg leader-followers model is not a satisfactory solution to the oligopoly problem, although it provides a legitimate criticism of the assumption that oligopolists follow reaction functions. Similarly, we are not claiming that under the GLOM one person would lead and the rest follow, but only that the assumption of competitive behavior implies that at least “See
Fellner (1949, ch. III) for an exposition
of Stackelberg’s
theory
of oligopoly.
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371
some individuals are behaving like ‘followers’ when they could have greater utility as ‘leaders’.r3 A case can be made that the Cournot solution does solve the problem of oligopoly, not because firms follow reaction functions, but because Cournot’s solution is a Nash equilibrium for the non-cooperative oligopoly game.r4 Similarly, GL assert that the equilibrium under their assumption of competitive behavior is a Nash equilibrium.r5 However, the analogy is misleading: if the oligopoly game is played only once, then, in the Nash equilibrium, firms immediately pick the intersection point of the Cournot reaction functions, but these strategies are not picked for Cournot’s reason, since reacting is not possible in a one-move game; for the same reason leading and following are also impossible, and Stackelberg’s criticism of Cournot is therefore irrelevant. However, the crucial difference between the oligopoly problem and the free-rider problem is that in the former every player knows all the possible pay-offs to himself and to others, while in the latter the pay-offs to others are not known. Therefore, in the oligopoly models for which the intersection of the Cournot reaction functions is also the equilibrium of the Nash non-cooperative game there is no tdtonnement process, but in the free-rider problem under the GLOM such a process is essential, since competitive behavior, in GL’s sense, is not based on game theory but involves behaving like a firm following a reaction function. Like the DRP, the DRG breaks down if there are income effects in the demand for the public good. A second limitation of the DRG is the requirement that the probability density functions of the parameters which determine preferences must be objectively known. Finally, it is important to remember that the mere existence of theoretical limitations to the efficiency of these three mechanisms is not necessarily inconsistent with the absence of any serious practical defects; however, the apparent absence of any examples of the real-world use of any of these mechanisms suggests that they do suffer from serious practical defects. “Groves and Ledyard are of course aware model [Groves and Ledyard (1977, p. 807)]. ‘%ee Friedman (1977, pp. 168-171). “See Groves and Ledyard (1977, p. 807).
that competitive
behavior
is not optimal
in their
References Arrow, K.J., 1977, The property rights doctrine and demand revelation under incomplete information, Institute for Mathematical Studies in the Social Sciences, Stanford University, Technical Report No. 243. Clarke, E.H., 1971, Multipart pricing of public goods, Public Choice 11, 17-33. Fellner, W.J., 1949, Competition among the few (A.A. Knopf, New York). Friedman, J.W., 1977, Oligopoly and the theory of games (North-Holland, Amsterdam). Groves, T., 1973, Incentivies in teams, Econometrica 41. 617-633. Groves, T. and J.O. Ledyard, 1977, Optimal allocation of public goods: A solution to the freerider problem, Econometrica 45, 7833810.
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Groves, T. and J.O. Ledyard, 1980, The existence of efficient and incentive compatible equilibria with public goods, Econometrica 48, 1487-1506. Groves, T. and M. Loeb, 1975, Incentives and public inputs, Journal of Public Economics 4, 21 l-226. Johnson, H.G., 1977, A note on the theory of public goods and ‘bluffing’, Journal of Public Economics 7, 3833386. Tideman, T.N. and G. Tullock, 1976, A new and superior process for making social choices, Journal of Political Economy 84, 1145-l 159. Vickrey, W., 1961, Counterspeculation, auctions and competitive sealed tenders, Journal of Finance 16. 8-37.