- Email: [email protected]
Social contract theory and ordinal distributive equity
Journa
of Public
Economics
SOCIAL
Elisha
5 (1976) 261-268.
Publishing
CONTRACT THEORY AND DISTRIBUTIVE EQUITY A. PAZNER Tel-Aviv
Received
0 North-Holland
April
and David
University,
1974, revised
ORDINAL
SCHMEIDLER*
Tel-Aviv,
version
Company
Israel
received
December
1974
The concept of the original position, implicit in some of Lerner’s and Harsanyi’s works, lies at the heart of Rawls’ contractarian approach to the theory of justice. This conceptual framework is used here to analyze the problem of distributive equity. It is shown that in a standard pure exchange economy, a social contract specifying the egalitarian distribution of economic resources might be expected to emerge in the original position. This result holds for any number of individuals and any number of commodities. The analysis does not involve interpersonal welfare comparisons and can be carried out within a purely ordinal framework of individual preferences.
1. Introduction The search for rules of social justice and distributive equity, a seemingly eternal conundrum of moral philosophy, has attracted the attention of normative economists for quite some time. Recurrent difficulties in the various attempts to incorporate such notions into economic analysis have to do with the cardinality of the individual utility functions used for such purposes and with the related problem of interpersonal comparisons of utilities whenever these are being made. These features are prominent throughout the classical utilitarian literature, are implicit in much of modern welfare economics and are in sharp contrast to the thoroughly ordinal nature of most of the economic theory of efficiency. In this paper, inspired by Rawls’ A Theory of Justice, we use the conceptual framework of social contract theory to show that the choice of distributive rules can be analyzed within an ordinal setting devoid of interpersonal comparisons of utilities; furthermore the analysis leads to determinate normative implications regarding distributive equity. The ordinal nature of the present discussion may be contrasted with the cardinality underlying the related works by Harsanyi (1953, 1955) and Lerner (1944) and the interpersonal comparisons *The research of both authors has been supported by a grant from the Ford Foundation. An earlier version of this paper appeared as working paper no. 27 (July, 1973) of the Foerder Institute for Economic Research at Tel-Aviv University. The authors wish to thank the Editor and an anonymous referee for their helpful suggestions.
262
E.A. Pazner
and D. Schmeidler,
Socral contract
theory
of utilities present in the latter. Regarding the theory of social choice pioneered by Arrow (1963), the unanimously agreed upon social contract (‘constitution’) emerging from the present analysis may be an indication that the contractarian approach might offer a way out of some of the disturbing difficulties raised by Arrow’s inquiry.
2. The contractarian framework The analysis here is Rawlsian in the sense that we are concerned with types of institutional choices that rational individuals may be expected to adopt in an ‘original position’ in which a ‘veil of ignorance’ may cover both their personal features (say preferences) and their placing in the socioeconomic system (in terms, say, of the particular bundle to be assigned to them in any given distribution of bundles). We depart however from Rawls in at least one important respect. Rawls’ fundamental moral principle that society ought to maximize the welfare of the least well off we understand as meaning that this (the maximin function) is the social welfare function which Rawls’ rational individuals are expected to accept ex ante (i.e. in the ‘original position’) as the guiding device in deciding about resource (bundles) allocation ex post (i.e. when individuals already know who they are). Without going into the question of whether the acceptance of this principle is as compelling as Rawls wishes to suggest [see Harsanyi (1973) for a critical examination of this issue], the operational meaning of this criterion in terms of the resulting distribution(s) of economic resources is dubious since it will depend on the particular (arbitrary?) way in which interpersonal comparisons of welfare (needed in order to determine who in any particular allocation is the worst off individual) will eventually have to be made [see also Arrow (1973)]. The point of view taken by us here differs from Rawls’ in that we seek to determine ex-ante a determinate distribution of economic resources (rather than an agreed upon social welfare function) on which rational individuals can be expected to agree unanimously in the original position. We show in the context of a standard pure-exchange economy that individuals in an ‘original position’ in which they are faced by distributions of bundles, without knowing the particular bundle to be assigned to them in any particular distribution, will always unanimously choose the egalitarian distribution independently of whether or not their preferences are known to them. It is also shown that the unanimous consensus on the egalitarian distribution of endowments is an extremely robust result. Indeed, it obtains for any of the standard criteria of decision making under complete ignorance (i.e. the maximin criterion, the minimax regret criterion and the principle of insufficient reason) which rational individuals may be expected to follow in the original position. The inherent risk-aversion underlying the above-mentioned decision criteria,
E.A. Panzer and D. Schmeidler, Social contract theory
263
coupled with the fundamental symmetry of the original position, turns out to dictate unequivocally a perfectly egalitarian outcome. As is well known, maximization of any symmetric strictly quasiconcave and monotone increasing function over a convex and symmetric domain leads to a symmetric solution. Applying this reasoning to the problem of distributing a fixed aggregate endowment of commodities across the members of the society, the egalitarian distribution is optimal under any ‘social welfare function’ satisfying the above conditions (over the domain of individual consumption bundles). This is easily seen to be the link explaining the identical results obtained under the seemingly different approaches considered here. In addition, it also turns out that both the maximin and the minimax regret criteria are ordinal in the present problem as the egalitarian distribution turns out to be invariant under any positive monotone transformations of the utility function(s) when either of these decision criteria is followed. This invariance also implies the absence of interpersonal comparisons of utilities, an important result. It is also worthwhile emphasizing that the egalitarian distribution will be agreed upon unanimously even in the case where the individuals have ‘vested interests’ in the original position in terms of knowing already then what their precise utility function is; the fact that the bundles are not named in the original position turns out to be enough for the egalitarian result. Finally, we wish to emphasize that the analysis is conducted throughout within the framework of timeless pure exchange economies in which distributional issues arise in their most primitive form. The question of possible extensions of the approach to production economies and to problems of resource distribution over time will be mentioned in the final section of the paper. 3. The egalitarian social contract In the ‘original position’ suppose that there are H individuals who have to decide unanimously on the initial distribution of whatever aggregate economic resources the society may have at its disposal. The distinctive feature of the social contract under consideration lies in its informational structure as it is assumed that what any particular individual will actually get in the decided distribution cannot be known to him at the time of signing of the contract. Consider first the case where each individual knows his (economic) preferences. To be more specific, assume that there are N commodities in our exchange economy, the aggregate endowment being a vector, say LO,in RN, -the nonnegative orthant of the N-dimensional Euclidean space-and the preferences of the hth individual (h = 1, . . ., H) (assumed here to be known by the individual) are given by a continuous strictly quasiconcave, monotone increasing function, say u,,, from RN, to the real numbers. A distribution of w is an H-list of bundles x = (x1, . . . , xH) in Ry s.t. cFS1 x,, = w; note that x being a
264
E.A. Pazner and D. Schmeidler,
Social contract theory
distribution means that x,, is not (necessarily) the bundle assigned to individual 12. Indeed, the essence of this case lies in the assumption that each individual knows only that one bundle in x will accrue to him, but does not know in advance which bundle this is going to be.
3.1. The maximin
social contract
Assuming that maximin criterion
each individual 12 (h = 1, . . . , H) applies (separately) in ranking the distributions, it is readily seen that
the
Max Min u,,(x~,) = u,,(o/H), h’ X where x runs over all distributions and h’ = 1, . . . , H. In other words, the maximin criterion leads every individual to select the egalitarian distribution as his best choice. Using the terminology of Hurwicz (1973) we can say that the application of the maximin criterion under the informational assumptions stated above gives rise to a mechanism that assigns equal endowments to every individual in the society. Note further that while we assumed each individual to know his u,,, all he has to know actually is that his u,, is one of H given utility functions, each one of which will eventually accrue to one of the H individuals. We shall now prove that this is indeed so. The problem faced in the ‘original position’ by each individual is now to find a distribution for which Max Min Min u(x~,) 11’ X u is obtained, where x varies over all distributions, u runs over H given continuous, strictly quasiconcave and monotone increasing real valued functions on RN,, andh’= I,..., H. Suppose, by contradiction, that the maximinmin is obtained at the distribution y, utility uO and bundle h,, i.e. Max Min Min u(x,,,) = u,(y,,,) h’ X u
> Min u(w/H)
= q(o/H).
u
The left-hand side of the inequality is clearly smaller or equal than Min,, ur(y,,,). which is Thus we have for all h’ (h’ = 1, . . . , H) that IQ(‘~,,,) > ul(w/H), impossible since o/H is a Pareto optimal allocation for the economy in which all the individuals have the same utility, namely ul. It follows that the social contract will specify egalitarian endowments when the structure of information, as in Rawls, is such that in the original position the individuals are not only ignorant of their own placing in the social system but are ignorant of their personal features (preferences) as well.
E.A. Pazner and D. Schmeidler,
Social contract
theory
265
In other words, a true state of primordial equality, where each individual may in the full sense of the term by any of the members of the society for which the social contract has to be signed, implies unanimous agreement on the egalitarian distribution of resources when each individual insists on getting at least his maximin. We conclude the present discussion by noting that since the egalitarian distribution that results from the maxmin(min) procedure is invariant under any monotone positive transformations of the utility functions, the apparent presence of interpersonal comparisons of utility is misleading and the procedure is truly devoid of any such comparisons. 3.2. The minimax regret social contract One might also wish to consider the following problem in the original position. Suppose that all that the individuals know about their preferences is that they are going to be continuous, strictly quasiconcave and monotone increasing. The question then is whether application of the maximin criterion by each individual will again lead to a meaningful social contract regarding the distribution of economic resources. Formally, the optimization problem faced by each individual is now Max Min Min u(x,,), X U h’ where x varies as before on all distributions, u now runs over the entire class of continuous, strictly quasiconcave and monotone increasing utility functions, and h’ = 1, . . . , H as before. Noting that now Min, has to be replaced by Inf,, and that Inf Min u(xh,) = - cc U h’ independently of x, the distributive result is indeterminate in this case as maximization of this function will imply that each distribution is as desirable as any other under the maxinf(min) criterion. This shows that one has to place explicit informational restrictions on the structure of knowledge supposed to prevail in the original position in order for the maximin criterion to lead to a meaningful (determinate) social contract. We saw above that it is enough for each individual to be faced by H (in fact, any finite number of) utility functions one of which will eventually be his, in order for the (maximin) procedure to lead to a determinate (egalitarian) distribution. It seems however that the spirit of Rawls’ ‘veil of ignorance’ in the original position is more in line with the case where all the individual knows is that his preferences will be drawn from the entire above mentioned class of utility functions, rather than from an a priori given finite subset therefrom. In order to
266
EA.
Pazner and D. Schmeidler,
Social contract theory
solve this kind of problem, another criterion for decision-making under ignorance has to be applied since the maximin criterion leads to indeterminate results. This also brings us to a more fundamental question that can be addressed to Rawls’ analysis, namely why confine attention solely to the maximin criterion in the first place. In particular, Savage’s minimax regret criterion seems on a priori grounds to be no less plausible a criterion, except perhaps for the cardinal notions usually involved in it. As we shall show that in the specific context considered here cardinality is not really required in applying the minimax regret criterion, this certainly enhances the appeal of this criterion in the present context. Applying the minimax regret criterion, with regrets measured by the maximin procedure, the problem faced by each individual is to find the distribution where Min Sup [Max Min u&,) - Min u(+)] X ” h’ h’ Y obtains.
From our earlier discussion
we know that
Max Min u(y,,,) = u(w/H), Y
and that whenever u(o/H)
h’
x # o/H, > Min u(xhP). h’
Thus whenever x # o/H the regret is positive independently of u and the minimal regret, which is zero, is obtained at the egalitarian distribution o/H once more. Since u runs over the entire class of strictly quasiconcave, continuous and monotone increasing utility functions, there is no operational meaning to a possible claim that interpersonal comparisons of utilities are involved in the present use of this criterion, and the procedure is truly ordinal.
3.3, The insuf$cient reason social contract Another possible criterion, used in a similar context by Lerner (1944) and Harsanyi (1953, 1955), is that based on the principle of insufficient reason. Assuming first that each individual knows his strictly concave utility function u but does not know in advance the bundle xW to be allocated to him in the selected distribution, it is easily seen that the social contract would lead once more to the egalitarian distribution (if he assigns the equal probability l/H
E.A. Pazner and D. Schmeidler,
to each bundle). that
Social contract theory
This is so since the strict concavity
of u immediately
267
implies
The egalitarian result continues to hold (under the same, subjective, equiprobability assignment) if in addition the individual is to be assigned his utility function by a random process independent of the assignment of the bundle to him [see Sen (1973) and references therein for the case of one-dimensional commodity bundles]. However, an arbitrary monotone (continuous and increasing) transformation of the utility function(s) will not always preserve the egalitarian result (as strict concavity is required for the egalitarian result). This procedure is therefore cardinal (although interpersonal comparisons of utilities still need not be involved).
4. Concluding remarks (1) The egalitarian social contract shown here to arise spontaneously for any of the standard decision criteria under ignorance, which any rational individual might use in ranking social states, illustrates the power of the concept of the original position insofar as equity analysis is concerned. In the case of a pure exchange economy, where no one can have any prior claims on the social dividend, the analysis shows that it is possible to advocate the intuitively appealing egalitarian distribution of resources without having recourse to problematic interpersonal utility comparisons (and without having to give up the cherished ordinal framework of standard economic theory). It should, however, be stressed that the assumption of risk-aversion in the original position is essential for the egalitarian result. (2) The fact that the egalitarian distribution of resources will usually fail to be Pareto optimal can, in principle, easily be remedied. After the social contract has been signed, and hence the veil of ignorance lifted, individuals may be permitted to trade in competitive markets so as to ensure the absence of a social dead weight loss. As they start with equal endowments, any competitive equilibrium will also be fair (in the sense that each individual will prefer his competitive bundle to that of any other individual), since the problems raised in Pazner and Schmeidler (1974a) and Varian (1974) do not apply to the ‘classical environment’ of the pure exchange economy under consideration. Hence, the intuitively appealing criterion of fair and Pareto efficient allocations can also be rationalized in terms of some kind of sequential contractarian situation. But note that from the viewpoint of the original position proper, the notion of fairness (i.e. absence of envy) precedes any efficiency considerations. Also, turning to the viewpoint of incentive compatibility, while the competitive mechanism is not cheatproof
268
E.A. Pazner and D. Schmeidler, Social contract theory
[in the sense of Hurwicz (1973)] - and hence the efficiency result of the ‘sequential’ allocative mechanism is actually in doubt-note that the egalitarian social contract is clearly incentive compatible (from the viewpoint of each individual in the original position). Finally note that in the original position, individuals should not be made aware of the possibility of recontracting out of the egalitarian distribution via the competitive mechanism; for if they knew this the problem might well have no solution at all (i.e. no social contract might be reached). (3) Finally, we mention that the approach presented here can be used in the case of production economies. However, while it is possible to show that a variant of the egalitarian result obtains in this case as well (in the sense that each individual will be given a bundle lying on the boundary of an ‘egalitarian’ production set) the efficiency problems become acute indeed. For, depending upon the veil of ignorance one might want to impose, it may even be impossible to define eficient production plans (e.g. when it is not known in the original position which utility functions are associated with which particular labor service). It might however be worth mentioning that in the context of the problem of resource allocation over infinite time we have been able to show (for rational individuals in an original position in which they are left ignorant of the particular generation in which they are going to live) that while maximin behavior is inefficient, minimax regret consumption plans are consistent with the golden rule of saving [see Pazner and Schmeidler (1974b)].
References Arrow, K.J., 1963, Social choice and individual values, 2nd edition (Wiley, New York). Arrow, K.J., 1973, Some ordinalist-utilitarian notes on Rawls’s theory of .iustice, Journal of Philosophy 70,245-263. Harsanyi, J.C., 1953, Cardinal utility in welfare economics and in the theory of risk taking, Journal of Political Economy 61,434-435. Harsanyi, J.C., 1955, Cardinal welfare, individualistic ethics and interpersonal comparison of utility, Journal of Political Economy 63,309.321. Harsanyi, J.C., 1973, Can the maximin principle serve as a basis for morality? A critique of John Rawls’s theory, Working paper no. cP-351 (Center for Research in Management Science, University of California, Berkeley). Hurwicz, L., 1973, The design of mechanisms for resource allocation, The American Economic Review 63.1-30. Lerner, A.P., 1944, The economics of control (Macmillan, New York). Pazner. E.A. and D. Schmeidler. 1974a. A difficulty in the concept of fairness, The Review of Economic Studies 41,441-441. ’ Pazner, E.A. and D. Schmeidler, 1974b, Just saving and the golden rule, Working paper no. 38 (The Foerder Institute for Economic Research, Tel-Aviv University). Rawls, J., 1971, A theory of justice (Harvard University Press, Cambridge, MA). Sen, A.K., 1973, On ignorance and equal distribution, American Economic Review 63, 10221024. Varian, H.L., 1974, Equity, envy, and efficiency, Journal of Economic Theory 9,63-91.
of Public
Economics
SOCIAL
Elisha
5 (1976) 261-268.
Publishing
CONTRACT THEORY AND DISTRIBUTIVE EQUITY A. PAZNER Tel-Aviv
Received
0 North-Holland
April
and David
University,
1974, revised
ORDINAL
SCHMEIDLER*
Tel-Aviv,
version
Company
Israel
received
December
1974
The concept of the original position, implicit in some of Lerner’s and Harsanyi’s works, lies at the heart of Rawls’ contractarian approach to the theory of justice. This conceptual framework is used here to analyze the problem of distributive equity. It is shown that in a standard pure exchange economy, a social contract specifying the egalitarian distribution of economic resources might be expected to emerge in the original position. This result holds for any number of individuals and any number of commodities. The analysis does not involve interpersonal welfare comparisons and can be carried out within a purely ordinal framework of individual preferences.
1. Introduction The search for rules of social justice and distributive equity, a seemingly eternal conundrum of moral philosophy, has attracted the attention of normative economists for quite some time. Recurrent difficulties in the various attempts to incorporate such notions into economic analysis have to do with the cardinality of the individual utility functions used for such purposes and with the related problem of interpersonal comparisons of utilities whenever these are being made. These features are prominent throughout the classical utilitarian literature, are implicit in much of modern welfare economics and are in sharp contrast to the thoroughly ordinal nature of most of the economic theory of efficiency. In this paper, inspired by Rawls’ A Theory of Justice, we use the conceptual framework of social contract theory to show that the choice of distributive rules can be analyzed within an ordinal setting devoid of interpersonal comparisons of utilities; furthermore the analysis leads to determinate normative implications regarding distributive equity. The ordinal nature of the present discussion may be contrasted with the cardinality underlying the related works by Harsanyi (1953, 1955) and Lerner (1944) and the interpersonal comparisons *The research of both authors has been supported by a grant from the Ford Foundation. An earlier version of this paper appeared as working paper no. 27 (July, 1973) of the Foerder Institute for Economic Research at Tel-Aviv University. The authors wish to thank the Editor and an anonymous referee for their helpful suggestions.
262
E.A. Pazner
and D. Schmeidler,
Socral contract
theory
of utilities present in the latter. Regarding the theory of social choice pioneered by Arrow (1963), the unanimously agreed upon social contract (‘constitution’) emerging from the present analysis may be an indication that the contractarian approach might offer a way out of some of the disturbing difficulties raised by Arrow’s inquiry.
2. The contractarian framework The analysis here is Rawlsian in the sense that we are concerned with types of institutional choices that rational individuals may be expected to adopt in an ‘original position’ in which a ‘veil of ignorance’ may cover both their personal features (say preferences) and their placing in the socioeconomic system (in terms, say, of the particular bundle to be assigned to them in any given distribution of bundles). We depart however from Rawls in at least one important respect. Rawls’ fundamental moral principle that society ought to maximize the welfare of the least well off we understand as meaning that this (the maximin function) is the social welfare function which Rawls’ rational individuals are expected to accept ex ante (i.e. in the ‘original position’) as the guiding device in deciding about resource (bundles) allocation ex post (i.e. when individuals already know who they are). Without going into the question of whether the acceptance of this principle is as compelling as Rawls wishes to suggest [see Harsanyi (1973) for a critical examination of this issue], the operational meaning of this criterion in terms of the resulting distribution(s) of economic resources is dubious since it will depend on the particular (arbitrary?) way in which interpersonal comparisons of welfare (needed in order to determine who in any particular allocation is the worst off individual) will eventually have to be made [see also Arrow (1973)]. The point of view taken by us here differs from Rawls’ in that we seek to determine ex-ante a determinate distribution of economic resources (rather than an agreed upon social welfare function) on which rational individuals can be expected to agree unanimously in the original position. We show in the context of a standard pure-exchange economy that individuals in an ‘original position’ in which they are faced by distributions of bundles, without knowing the particular bundle to be assigned to them in any particular distribution, will always unanimously choose the egalitarian distribution independently of whether or not their preferences are known to them. It is also shown that the unanimous consensus on the egalitarian distribution of endowments is an extremely robust result. Indeed, it obtains for any of the standard criteria of decision making under complete ignorance (i.e. the maximin criterion, the minimax regret criterion and the principle of insufficient reason) which rational individuals may be expected to follow in the original position. The inherent risk-aversion underlying the above-mentioned decision criteria,
E.A. Panzer and D. Schmeidler, Social contract theory
263
coupled with the fundamental symmetry of the original position, turns out to dictate unequivocally a perfectly egalitarian outcome. As is well known, maximization of any symmetric strictly quasiconcave and monotone increasing function over a convex and symmetric domain leads to a symmetric solution. Applying this reasoning to the problem of distributing a fixed aggregate endowment of commodities across the members of the society, the egalitarian distribution is optimal under any ‘social welfare function’ satisfying the above conditions (over the domain of individual consumption bundles). This is easily seen to be the link explaining the identical results obtained under the seemingly different approaches considered here. In addition, it also turns out that both the maximin and the minimax regret criteria are ordinal in the present problem as the egalitarian distribution turns out to be invariant under any positive monotone transformations of the utility function(s) when either of these decision criteria is followed. This invariance also implies the absence of interpersonal comparisons of utilities, an important result. It is also worthwhile emphasizing that the egalitarian distribution will be agreed upon unanimously even in the case where the individuals have ‘vested interests’ in the original position in terms of knowing already then what their precise utility function is; the fact that the bundles are not named in the original position turns out to be enough for the egalitarian result. Finally, we wish to emphasize that the analysis is conducted throughout within the framework of timeless pure exchange economies in which distributional issues arise in their most primitive form. The question of possible extensions of the approach to production economies and to problems of resource distribution over time will be mentioned in the final section of the paper. 3. The egalitarian social contract In the ‘original position’ suppose that there are H individuals who have to decide unanimously on the initial distribution of whatever aggregate economic resources the society may have at its disposal. The distinctive feature of the social contract under consideration lies in its informational structure as it is assumed that what any particular individual will actually get in the decided distribution cannot be known to him at the time of signing of the contract. Consider first the case where each individual knows his (economic) preferences. To be more specific, assume that there are N commodities in our exchange economy, the aggregate endowment being a vector, say LO,in RN, -the nonnegative orthant of the N-dimensional Euclidean space-and the preferences of the hth individual (h = 1, . . ., H) (assumed here to be known by the individual) are given by a continuous strictly quasiconcave, monotone increasing function, say u,,, from RN, to the real numbers. A distribution of w is an H-list of bundles x = (x1, . . . , xH) in Ry s.t. cFS1 x,, = w; note that x being a
264
E.A. Pazner and D. Schmeidler,
Social contract theory
distribution means that x,, is not (necessarily) the bundle assigned to individual 12. Indeed, the essence of this case lies in the assumption that each individual knows only that one bundle in x will accrue to him, but does not know in advance which bundle this is going to be.
3.1. The maximin
social contract
Assuming that maximin criterion
each individual 12 (h = 1, . . . , H) applies (separately) in ranking the distributions, it is readily seen that
the
Max Min u,,(x~,) = u,,(o/H), h’ X where x runs over all distributions and h’ = 1, . . . , H. In other words, the maximin criterion leads every individual to select the egalitarian distribution as his best choice. Using the terminology of Hurwicz (1973) we can say that the application of the maximin criterion under the informational assumptions stated above gives rise to a mechanism that assigns equal endowments to every individual in the society. Note further that while we assumed each individual to know his u,,, all he has to know actually is that his u,, is one of H given utility functions, each one of which will eventually accrue to one of the H individuals. We shall now prove that this is indeed so. The problem faced in the ‘original position’ by each individual is now to find a distribution for which Max Min Min u(x~,) 11’ X u is obtained, where x varies over all distributions, u runs over H given continuous, strictly quasiconcave and monotone increasing real valued functions on RN,, andh’= I,..., H. Suppose, by contradiction, that the maximinmin is obtained at the distribution y, utility uO and bundle h,, i.e. Max Min Min u(x,,,) = u,(y,,,) h’ X u
> Min u(w/H)
= q(o/H).
u
The left-hand side of the inequality is clearly smaller or equal than Min,, ur(y,,,). which is Thus we have for all h’ (h’ = 1, . . . , H) that IQ(‘~,,,) > ul(w/H), impossible since o/H is a Pareto optimal allocation for the economy in which all the individuals have the same utility, namely ul. It follows that the social contract will specify egalitarian endowments when the structure of information, as in Rawls, is such that in the original position the individuals are not only ignorant of their own placing in the social system but are ignorant of their personal features (preferences) as well.
E.A. Pazner and D. Schmeidler,
Social contract
theory
265
In other words, a true state of primordial equality, where each individual may in the full sense of the term by any of the members of the society for which the social contract has to be signed, implies unanimous agreement on the egalitarian distribution of resources when each individual insists on getting at least his maximin. We conclude the present discussion by noting that since the egalitarian distribution that results from the maxmin(min) procedure is invariant under any monotone positive transformations of the utility functions, the apparent presence of interpersonal comparisons of utility is misleading and the procedure is truly devoid of any such comparisons. 3.2. The minimax regret social contract One might also wish to consider the following problem in the original position. Suppose that all that the individuals know about their preferences is that they are going to be continuous, strictly quasiconcave and monotone increasing. The question then is whether application of the maximin criterion by each individual will again lead to a meaningful social contract regarding the distribution of economic resources. Formally, the optimization problem faced by each individual is now Max Min Min u(x,,), X U h’ where x varies as before on all distributions, u now runs over the entire class of continuous, strictly quasiconcave and monotone increasing utility functions, and h’ = 1, . . . , H as before. Noting that now Min, has to be replaced by Inf,, and that Inf Min u(xh,) = - cc U h’ independently of x, the distributive result is indeterminate in this case as maximization of this function will imply that each distribution is as desirable as any other under the maxinf(min) criterion. This shows that one has to place explicit informational restrictions on the structure of knowledge supposed to prevail in the original position in order for the maximin criterion to lead to a meaningful (determinate) social contract. We saw above that it is enough for each individual to be faced by H (in fact, any finite number of) utility functions one of which will eventually be his, in order for the (maximin) procedure to lead to a determinate (egalitarian) distribution. It seems however that the spirit of Rawls’ ‘veil of ignorance’ in the original position is more in line with the case where all the individual knows is that his preferences will be drawn from the entire above mentioned class of utility functions, rather than from an a priori given finite subset therefrom. In order to
266
EA.
Pazner and D. Schmeidler,
Social contract theory
solve this kind of problem, another criterion for decision-making under ignorance has to be applied since the maximin criterion leads to indeterminate results. This also brings us to a more fundamental question that can be addressed to Rawls’ analysis, namely why confine attention solely to the maximin criterion in the first place. In particular, Savage’s minimax regret criterion seems on a priori grounds to be no less plausible a criterion, except perhaps for the cardinal notions usually involved in it. As we shall show that in the specific context considered here cardinality is not really required in applying the minimax regret criterion, this certainly enhances the appeal of this criterion in the present context. Applying the minimax regret criterion, with regrets measured by the maximin procedure, the problem faced by each individual is to find the distribution where Min Sup [Max Min u&,) - Min u(+)] X ” h’ h’ Y obtains.
From our earlier discussion
we know that
Max Min u(y,,,) = u(w/H), Y
and that whenever u(o/H)
h’
x # o/H, > Min u(xhP). h’
Thus whenever x # o/H the regret is positive independently of u and the minimal regret, which is zero, is obtained at the egalitarian distribution o/H once more. Since u runs over the entire class of strictly quasiconcave, continuous and monotone increasing utility functions, there is no operational meaning to a possible claim that interpersonal comparisons of utilities are involved in the present use of this criterion, and the procedure is truly ordinal.
3.3, The insuf$cient reason social contract Another possible criterion, used in a similar context by Lerner (1944) and Harsanyi (1953, 1955), is that based on the principle of insufficient reason. Assuming first that each individual knows his strictly concave utility function u but does not know in advance the bundle xW to be allocated to him in the selected distribution, it is easily seen that the social contract would lead once more to the egalitarian distribution (if he assigns the equal probability l/H
E.A. Pazner and D. Schmeidler,
to each bundle). that
Social contract theory
This is so since the strict concavity
of u immediately
267
implies
The egalitarian result continues to hold (under the same, subjective, equiprobability assignment) if in addition the individual is to be assigned his utility function by a random process independent of the assignment of the bundle to him [see Sen (1973) and references therein for the case of one-dimensional commodity bundles]. However, an arbitrary monotone (continuous and increasing) transformation of the utility function(s) will not always preserve the egalitarian result (as strict concavity is required for the egalitarian result). This procedure is therefore cardinal (although interpersonal comparisons of utilities still need not be involved).
4. Concluding remarks (1) The egalitarian social contract shown here to arise spontaneously for any of the standard decision criteria under ignorance, which any rational individual might use in ranking social states, illustrates the power of the concept of the original position insofar as equity analysis is concerned. In the case of a pure exchange economy, where no one can have any prior claims on the social dividend, the analysis shows that it is possible to advocate the intuitively appealing egalitarian distribution of resources without having recourse to problematic interpersonal utility comparisons (and without having to give up the cherished ordinal framework of standard economic theory). It should, however, be stressed that the assumption of risk-aversion in the original position is essential for the egalitarian result. (2) The fact that the egalitarian distribution of resources will usually fail to be Pareto optimal can, in principle, easily be remedied. After the social contract has been signed, and hence the veil of ignorance lifted, individuals may be permitted to trade in competitive markets so as to ensure the absence of a social dead weight loss. As they start with equal endowments, any competitive equilibrium will also be fair (in the sense that each individual will prefer his competitive bundle to that of any other individual), since the problems raised in Pazner and Schmeidler (1974a) and Varian (1974) do not apply to the ‘classical environment’ of the pure exchange economy under consideration. Hence, the intuitively appealing criterion of fair and Pareto efficient allocations can also be rationalized in terms of some kind of sequential contractarian situation. But note that from the viewpoint of the original position proper, the notion of fairness (i.e. absence of envy) precedes any efficiency considerations. Also, turning to the viewpoint of incentive compatibility, while the competitive mechanism is not cheatproof
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[in the sense of Hurwicz (1973)] - and hence the efficiency result of the ‘sequential’ allocative mechanism is actually in doubt-note that the egalitarian social contract is clearly incentive compatible (from the viewpoint of each individual in the original position). Finally note that in the original position, individuals should not be made aware of the possibility of recontracting out of the egalitarian distribution via the competitive mechanism; for if they knew this the problem might well have no solution at all (i.e. no social contract might be reached). (3) Finally, we mention that the approach presented here can be used in the case of production economies. However, while it is possible to show that a variant of the egalitarian result obtains in this case as well (in the sense that each individual will be given a bundle lying on the boundary of an ‘egalitarian’ production set) the efficiency problems become acute indeed. For, depending upon the veil of ignorance one might want to impose, it may even be impossible to define eficient production plans (e.g. when it is not known in the original position which utility functions are associated with which particular labor service). It might however be worth mentioning that in the context of the problem of resource allocation over infinite time we have been able to show (for rational individuals in an original position in which they are left ignorant of the particular generation in which they are going to live) that while maximin behavior is inefficient, minimax regret consumption plans are consistent with the golden rule of saving [see Pazner and Schmeidler (1974b)].
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