Uniform externalities: Two axioms for fair allocation

Journal of Public Economics 43 (1990) 3055326. North-Holland

UNIFORM EXTERNALITIES

Two Axioms for Fair Allocation

Herve MOULIN*

Duke University, Durham, NC27706, USA

Received May 1989, revised version received March 1990

Positive (resp. negative) Preference Externalities say that an agent always prefers his actual welfare to his (virtual) welfare should other agents share his preferences (resp. prefers his virtual to his actual welfare). Negative Group Externalities say that an agent never prefers his actual welfare to his (virtual) welfare should he be the sole user of the resources.

These two axioms unify several familiar fairness properties ~ and yield some new ideas as well _ in the division of unproduced commodities and the cooperative production of a private or a public good. We also discuss their compatibility with No Envy and Resource Monotonicity.

1. The utilization of common property resources

A seminal question of distributive justice consists of allocating commonly owned resources among a given set of agents. By resources we mean a set of vectors of goods. A solution must choose one of those vectors, and distribute it among individual agents. For instance, in the classical fair division problem [Steinhaus (1948), Varian (1974)] there is a single such vector of goods. Alternatively, in a cooperative production problem, the resources are the production technology, and an allocation is a feasible production plan specifying input contributions and output shares. To complicate things a bit, the output may be a public good.

Examples of common property resources include broadcasting frequencies (an unproduced commodity), manganese nodules on the deep sea bed (here the mining technology is the resource), the cost sharing of public radio (a pure public good) as well as the pricing of public utilities (here the resource is the increasing returns technology for providing the utility).

We assume, in the tradition of the (once) ‘new’ welfare economics, that

*Research supported by NSF Grant SES8618600 and by the grant PB86-0613 from the Direction General de la Investigation Cientifica y Tecnica, Spanish Ministry of Education. Stimulating discussions with J. Cremer, and W. Thomson are acknowledged. Special thanks to J. Roemer and to an anonymous referee for their criticisms.

004772727/90/$03.50 0 199tLElsevier Science Publishers B.V. (North-Holland)

306 H. Moulin, Uniform externalities

agents are endowed with ordinal, non-intercomparable preferences.’ As the agents have equal claims on the resources, the only source of disagreement is the difference of individual preferences. Should they all have the same preferences, the compellingly equitable allocation of resources yields the highest feasible equal utility outcome (equal treatment of equals). If, however, their preferences differ, what do we mean by fair allocation?

In this paper we propose two axioms that appear to be helpful in a variety of fair allocation problems, including the cooperative production of a private or a public good, and the division of unproduced commodities. These two axioms place an upper or a lower bound on every agent’s utility, and these bounds depend exclusively upon this agent’s preferences and the feasibility constraints (but they do not depend upon other agents’ preferences). Thus, they play the role respectively of an individual rationality constraint (the case of a lower bound) and of an a priori cap on individual gains (the case of an upper bound). Interestingly, each of one of our two axioms (called Uniform Preference Externalities and Uniform Group Externalities, in short UPE and UGE) may yield an upper bound or a lower bound, depending on the particular problem where we use it: for instance, in cooperative production of a private good, the UPE axiom yields a lower bound in the case of a decreasing returns process, and an upper bound if the production process has increasing returns (see section 5 below).

Our first axiom, Uniform Preference Externalities, conveys the idea that

differences in preferences create an externality for which all agents are jointly responsible, hence this externality should be uniformly borne by the agents (be it a positive or a negative externality). Of course, in the absence of interpersonal comparison of utilities, we cannot make agent 1 and agent 2 experience the same variation of utility from the externality, but we may at least make sure that the sign of these variations is the same: they both suffer or they both are hurt.

Given an allocation problem and a particular solution (allocation mechanism), every agent makes the following thought experiment: what happens if every other agent had precisely the same preference as mine? Under equal treatment of equals, the solution picks the highest equal utility outcome. Call this level the unanimity utility of our agent. When his actual utility (in the true economy where different agents have different preferences) exceeds his (virtual) unanimity utility, our agent enjoys a positive prqference externality (he is glad to disagree with the others); if actual utility is below unanimity utility, he enjoys a negative preference externality. The principle of Uniform

‘This is in line with a large portion of the economic literature on distributive justice. See, in particular, the concepts of No Envy and resource monotonicity discussed in section 7 iAuK. However, experimental evidence demonstrates the importance of interpersonal comparison If needs and/or welfare [see, for example, Schokkaert and Overlaet (1983) and Ynari and Bar- Hillel (1984)]; and institutional examples of differences in claims abocnd

H. Moulin, Uniform externalities 307

Preference Externalities (UPE) rules out the possibility that in a given economy the preference externality of one agent is positive while that of another agent is negative.

As our first example, consider the fair division of a lixed vector of divisible goods. If we all have the same tastes, fairness demands we split those goods equally (except possibly when preferences are not convex); thus the Uniform Preference Externalities axiom sets the equal split utility as a lower bound on any agent’s welfare: everyone should get at least a fair share (in his view) of the pie. In other words, everyone enjoys a positive (or zero) preference externality. Of course this lower bound is as old as the fair division problem itself [see Steinhaus (1948)]. By contrast, in our next example, the UPE axiom uncovers a new idea.

In the provision of a public good problem, the agents must choose the level of production of the public good and share its cost. If they all have the same preferences, the fair outcome is obvious: every agent pays the average (per capita) cost; given uniform cost-sharing, the identical demand of every agent is met. Hence the overall preference externality is negative: because their preferences differ, the individual demands (under uniform cost-sharing) do not coincide, thus it is in general not possible to guarantee the unanimity utility to every agent. Consequently the Uniform Preference Externalities axiom takes the unanimity utility as an upper bound on individual welfares: nobody should prefer his actual allocation to the satisfaction of his demand for the public good when he pays average cost. This upper bound has a lot of bite: it rules out familiar cost-sharing methods such as the Lindahl equilibrium [and its generalization as Kaneko’s ratio equilibrium - see Kaneko (1977)] and the egalitarian equivalent method proposed by Moulin (1987). We state this upper bound precisely in section 3 but refer the reader to Moulin (I989b) for a detailed discussion of its consequences.

The UPE axiom is meaningful only if it makes sense to exchange preferences across agents: What if you had my preferences? In turn, this means that everyone is responsible for his own tastes; we rule out the interpretation of those preferences as needs: on this, see the discussion in Moulin (1989b) as well as the example in section 5 below.

Our second axiom, Uniform Group Externalities, rests on another thought experiment, feasible in any resource allocation problem: What happens if I am alone to utilize those resources? Call the corresponding utility level the free access utility of this agent. If his actual utility (in the true economy) exceeds his free access utility, we say that he enjoys a positive group

externality. If the actual utility is below the free access utility, he enjoys a negative group externality. The principle of Uniform Group Externalities

(UGE) says that if an agent enjoys a positive group externality, no other agent can enjoy a negative group externality.

In the provision of a public good, the group externality is clearly positive:


more agents do not reduce my consumption of the public good and they will pay some share of its cost. The free access utility is achieved when the agent can choose the level of the public good provided he covers its full cost. Thus the Uniform Group Externalities axiom yields the individual rationality constraint originally introduced by Foley (1967): every agent has free access to the technology, so an agent’s utility is at least his free access utility (or he could start to produce the good on his own).

In the fair division of (unproduced) commodities, the group externality is negative: the more we are to eat the pie the smaller the average share. Here the free access utility is that from eating the whole pie, so that the Uniform Group Externalities principle is met by every mechanism. There are many other problems where the group externality is negative, and the free access upper bound has bite: an example is the cooperative production where the technology has decreasing returns to scale [see section 5 and Moulin (199Oc)J

The rest of the paper is organized as follows. In section 2 we define our allocation problems and the notion of a solution. Section 3 is devoted to Uniform Preference Externalities, and section 4 to Uniform Group Externali- ties. Section 5 discusses these two axioms in the particularly interesting model of cooperative production of a private good. In section 6 we examine the compatibility of our two Uniform Externalities axioms. Finally, section 7 compares our two axioms with two competing interpretations of fairness, namely the No Envy [Foley (1967), Varian (1974)] and the Resource Monotonicity [Moulin (1988a), Moulin and Thomson (1988) Roemer (1986)] properties. We note that, for our sample of representative problems and within the class of Pareto-optimal solutions, No Envy is always compatible with Uniform Preference Externalities and is generally incompatible with Uniform Group Externalities. On the contrary, Resource Monoto- nicity is always compatible with Uniform Group Externalities and always incompatible with Uniform Preference Externalities. Several related papers by this author develop some of these observations into formal theorems [Moulin (1989a, 1989b, 1990a)]. The current paper contains no technical proof.

2. The model

To illustrate the notion of an allocation problem, we use six representative problems.

Problems I and 2: Division of Goods and Division of Buds [Thomson and Varian (1985)].

The vector of commodities w (a positive vector in RP) is to be divided among n agents in non-negative shares zr,. . . , z,:


Zi~Oall i and C zr=o. i=l

In Division of Goods, every agent has convex preferences, strictly increasing in each commodity. In Division of Bads, those preferences are convex and strictly decreasing in each commodity.

Problems 3 and 4: Provision of a Public Good and Provision of a Public Bad

[Diamantaras (1987), Moulin (1987)].

Start with the public good model. Our n agents must pick the level a, az0,

at which the public good will be produced, and they must share its cost c(a), c(a) 2 0, in some equitable way. If agent i consumes a units of the public good and pays ci (in dollars), his final utility is ui(a,ci).

The utility is increasing in a and strictly decreasing in ci. In the Public Bad case, the input is now a desirable commodity and the

output is a bad (e.g. pollution) that all agents would like to see as low as possible. So the function ui is decreasing in a and strictly increasing in ci: c(a)

can be interpreted as the amount of private benefits allowed by the pollution level, a. Alternatively, the input ci measures agent i’s activity level and the inverse of the function c is the production function of the public bad output.

Problems 5 and 6: Cooperative Production with DRS, Cooperative Production

with IRS [Mirrlees (1974) Moulin (1990a), Moulin and Roemer (1989) Roemer (1985, 1986)].

The n agents together use a single production process transforming one input into an output according to the function y=_/‘(x). The agents must choose a feasible production plan specifying the (non-negative) input contributions and (non-negative) output shares of every agent. Thus, a feasible allocation (zi,...,z,) is

zi=(xi,yi)20, for all i; jl ~i=/(?jll xi).

Agent i’s preferences are convex, and represented by a utility function ui(xi, yi) decreasing in xi and increasing in yi. In the DRS (decreasing returns to scale) model, the function f is non-decreasing concave and f(0) = 0. In the IRS (increasing returns to scale) model, this function is non-decreasing, convex and f(0) =O.


We now define, in the abstract, an allocation mechanism. Even though in the actual society individual preferences typically differ, potentially each agent is drawn from the same pool; in other words, the domain where individual preferences vary is the same for all agents. We write this domain as D. For every economy (i.e. for every preference profile), the feasibility constraints determine a subset of feasible utility vectors. A mechanism S associates to every economy (u,, . . , un) a Pareto optimal utility vector S(u, ,..., u,). Thus, Si(ul,. . .,un) is agent i’s utility at some Pareto-optimal allocation. We also assume that our mechanism is anonymous: exchanging agent i and agent j’s preferences will exchange these agents’ utilities without affecting any other agent’s utility.

Here is an important consequence of the Pareto optimality and anonymity assumptions. Suppose the agents in the economy are unanimous in their preferences: they all share the same utility function, uO. Then any (Pareto- optimal and anonymous) mechanism selects in this economy the highest feasible equal utility vector.* We call this level the unanimity level and write it un(u,; n):

for all i = 1,. . , n: Si(uo, uo, . . . , uo) = un (u,; n).

The unanimity level is independent of our particular choice of a mechanism. It plays a key role in the definition of our first axiom.

3. The axiom: Uniform preference externalities

Suppose an agent i knows his own preferences, ui, and society’s size, but ignores other agents’ preferences. We may call this state of information the thin veil of’ i,gnorance.3 Because all other agents have the same preference domain, D, it is conceivable that they would all have exactly the same preferences as he does. He would then enjoy his unanimity level un (ui; n).

In the actual preference profile where agents do ha.ve different preferences, u1 ,..., unr each agent i can compare his actual utility, Si(u, ,..., u,,), with his unanimity level, un (ui; n). If Si is larger than (resp. smaller than) un (ui, n), the differences in individual preferences bring a positive externality (resp. a

*Provided that a Pareto-optimal equal utility vector exists. This will always be the case when the Pareto frontier has ‘no hole’. For the precise statement of this topological assumption, we refer to Moulin (1989a). The assumption holds true in all examples discussed in the current paper.

3Roemer (1985) invented the term, but gave it a different meaning: an agent knows his own preferences but not his endowments.


negative externality) on agent i, at this particular profile and for this

particular mechanism.

The UPE axiom. The preference externalities of no two agents should have opposite signs: for all profiles ur,. . . , u, in D”, and any two agents i, j:

Si(ul,. . . , u,) > un (ui; n) = Sj(ul,. , u,) 2 un (uj: n). (2)

In other words if one agent benefits from the difference in preferences, no one else should suffer from it. So, loosely speaking, the preference externality bears uniformly on every agent. Notice that property (2) is purely ordinal and uses no interpersonal comparison of welfare.

The UPE principle places a lower bound on individual welfare when the vector of unanimity utilities, u = (un (aI; n), , un (u,; n)), is below the Pareto frontier; it places an upper bound when this vector is above the Pareto frontier; and when the vector of unanimity utilities is on the Pareto frontier, it forces to choose it precisely. Note that these bounds, like any individual rationality constraint, are decentralized: the unanimity utility depends only upon agent i’s preferences, the number of his partners, and the feasibility constraints. Yet, to know if un (ui; n) constitutes a lower or an upper bound is not a decentralized piece of information. We need to find if the utility vector, w=(un(u,;n),..., un (u,; n)), is below or above the Pareto frontier.

Only in one case does the UPE axiom yield a decentralized lower (resp. upper) bound: when the vector w is always on or below (resp. on or above) the Pareto frontier.

Definition I. We say that a problem has a positive preference externality

(PPE) if for each profile (u,,. .., u,,) in D”, the vector (un (u,; n), . . , un (u,; n)) is on or below the Pareto frontier of the corresponding economy. In this case the axiom of Uniform Preference Externalities places the following decentralized lower bound on individual welfare:

In words, the differences in individual preferences always generate a positive externality, from which all agents can benefit.

The Division of Goods and the Division of Bads are two problems with a positive preference externality. Indeed, when all agents have the same conuex preferences, equal split of the resources o is the unique equal utility Pareto- optimal outcome. So, un (ui; n) = u,(w/n).

Now for an arbitrary preference profile. Splitting w equally among the agents is always feasible, whence the vector of unanimity utilities,

312 H. Mot&n, Uniform externalities

(u,(wln),..., u,(o/n)), is on or below the Pareto frontier. Thus, the UPE axiom amounts to the equal split lower bound: everyone should get at least his fair share of the pie (.Sizui(w/n) for all i). Recall that the early literature on fair division [from Steinhaus (1948) to Dubins and Spanier (1961)] took this as the very definition of fairness.4 See also Dworkin’s (1981) ‘starting gate equality’. Notice finally that the problem of allocating indivisible goods (and money) admits a similar lower bound [see Alkan et al. (1988), Gale (1960), Maskin (1987), Svensson (1983)].

Definition 2. We say that a problem has a negative preference externality if for each profile (u,, . . , u,J in D”, the vector (un(u,;n),. ..,un(u,;n)) is on or above the Pareto frontier of the corresponding economy. The UPE axiom then requires that every agent bears a non-negative share of this externality:

The Provision of a Public Good (and the Provision of a Public Bad) are two problems with a negative preference externality. To see this, first compute the unanimity level un(ui; n). By anonymity, in a unanimous economy we must split the cost in equal shares, and the highest equal utility level is worth

(3)

We claim that the vector (un(u,;n),. . ,un(u,;n)) is always on or beyond the Pareto frontier. Suppose, by way of contradiction that we find a public good level a* and some cost shares (c,, . . , c,) such that

Cl + . ..+c.=c(a*); ui(a*, ci) 2 un (ui; n), for all i, (4)

with at least one strict inequality. In view of (3) this implies ui(a*,ci)z

ui(a*,c(a*)/n), whence cizc(a*)/n for all i. Moreover, a strict inequality in (4) implies ci <: c(a*)/n and yields the desired contradiction.5

Thus, in the provision of a public good problem, the UPE axiom focuses attention upon uniform cost-sharing: no agent should ever prefer his

4But after the introduction of the notion of envy [Foley (1967)], the term ‘fair’ was promptly reinterpreted as the combination of No Envy and Pareto optimality. I prefer not to use the word in the context of a formal definition

‘The above proof is quite general. It applies to an arbitrary set of public decision outcomes (a could represent a vector of public goods, or a candidate in an election, or a combination of both). We do not need to assume that agents’ preferences are monotonic or continuous in a, However, to guarantee that the UPE axiom is compatible with Pareto optimality of S requires some topological assumption [see Diamantaras (1987), Moulin (1989b)].


allocation to the satisfaction of his demand for the public good when he pays average cost. This bound is quite sharp. At most profiles it is violated by the Lindahl equilibrium (and by its modification, like Kaneko’s ratio equilibrium to accommodate non-linear costs). At many profiles it is violated as well by the egalitarian equivalent cost-sharing proposed in Moulin (1987).6 See Moulin (1989b) for more discussion of the UPE upper bound in the provision of a public good problem.

Next we propose an alternative interpretation of the UPE axiom in those problems with either a positive preference externality, or with a negative preference externality. If we have a positive preference externality, the unanimity level un(ui;n) is the highest of all (decentralized) lower bound functions compatible with a (Pareto-optimal and anonymous) mechanism. To see this, take any mechanism S and consider some abstract decentralized lower bound function $(ui;n), namely

si(” r, . . , a,) 2 Ic/(ui; n), for all i, all ur , . , u,.

At the unanimous profile where all preferences are ui, we have Si(ui, . . . , ui) = un(u,; n), hence the function $ is bounded above by the unanimity utility function. Thus, the UPE axiom is equivalent to the requirement that the mechanism provides each agent with the highest feasible guaranteed utility level compatible with equal treatment of equals. This is, in itself, an appealing justification of the UPE axiom, much in the spirit of the Lockean principle of maximal individual liberties compatible with similar liberties for all.

Similarly, in a problem with negative preference externality, the unanimity level, un(ui; n), is the lowest of all (decentralized) upper bound functions compatible with a (Pareto-optimal and anonymous) mechanism. Thus, a justification of inequalities UPE as the lowest feasible upper bound: every other agent wants to make sure that I cannot be ‘too’ lucky. See Moulin (1988b, 1989a).

To conclude this section, we emphasize that not every problem has a positive preference externality or a negative preference externality. For

‘A Lindahl-Kaneko equilibrium is characterized by a convex combination of ratios, rl,. ..,r, (~~20, c,ri= l), such that every agent i demands the same level a* of public good if his cost-share is r,:

ci = r, c(a*); q(a*. c,) = max ui(a. r;.r(a)). 020

If the ratios Y],.. .,r, all coincide (ri= l/n), then the LK equilibrium meets the unanimity upper bound (any agent’s L.K equilibrium utility is precisely his unanimity utility). In euery other case, at least one ratio r, is below l/n, whence agent i’s equilibrium utility exceeds un(u,;n). Turning to the egalitarian-equivalent allocation [Moulin (1987)J, even when the unanimity utility vector is Pareto optimal (and equals the LK utility vector), the egalitarian-equivalence mechanism may recommend quite a different outcome.


instance, consider the problem of dividing a vector of goods when preferences are monotonic but not necessarily convex. In such problems, the UPE axiom brings an upper bound at some profiles and a lower bound at other profiles. Another example is the model of cooperative production of a private good, when the returns of the technology first increase and then decrease. Of course, this complication reduces significantly the appeal of the UPE axiom.

4. The axiom: Uniform group externalities

Suppose agent i, endowed wth utility ui, is the sole user of the resources (namely unproduced commodities, or a technology to produce a private or a public good). We call the level un(ui; 1) that he would then reach by maximizing his utility over the set of feasible allocations the free access utility.

In the actual preference profile, ur, . . . , u,, each agent i compares his actual utility, Si(ul,. ., u,,), with his free access level, un(u,; 1). If Si is larger (resp. smaller) than un(ui: l), we say that agent i enjoys a positive (resp. negative) group externality, at this particular profile and for this particular mechanism.

The UGE axiom. The group externalities of no two agents should have opposite signs: for all profile u1 , . . . , u, in D” and any two agents i, j:

The UGE axiom requires the group externality to bear uniformly on every agent. Just like the UPE axiom, it is purely ordinal and uses no interpersonal comparison of preferences. Another feature common to both of these axioms is to place a lower bound or an upper bound on individual welfare, depending upon the position of a reference vector (in this case the vector of free access utilities, otherwise the vector of unanimity utilities) with respect to the Pareto frontier. Moreover, these bounds are especially appealing in those problems where the reference vector is always on the same side of the Pareto frontier.

Definition 3. We say that a problem has a negative group externality (resp. a positive group externality) if for every profile ur,. ,u, in D”, the vector (un (u,; l), . . , un(u,; 1)) is on or above (resp. on or below) the Pareto frontier of the corresponding economy. In this case the UGE axiom places the following decentralized upper bound (resp. lower bound) on individual

welfare:

si(“I>...> U,) s Un (Ui; 1) [resp. Si(ul,. . , un) 2 un (ui; l)].


The general formulation of the UGE axiom is new. Most of its applications of which I am aware, are not. The same comment of course applies to the UPE axiom.

The Division of Goods [problem (I)] is the simplest example of a problem with negative group externality, As un (ui; 1) = ui(w), the UGE axiom amounts to the upper bound Sisui(o) which is automatically satisfied.7 A more interesting example of negative group externality is the Provision of a Public Bad [problem (4)]. At a feasible allocation (a, ci, . . . , cn), the amount a of bad is produced and agent i enjoys the benefit (or activity level) ci:

for all i, c,zO and i ci=c(a). i=l

Hence,

ui(a, ci) 5 ~,(a, c(u)) 2 max ui(b, c(b)) = un (ui; 1). b,O

It is easy to see why the UGE upper bound can have a lot of bite, for instance if an agent dislikes the public bad very much relative to the private input.

Turning now to problems with positive group externalities, our first (trivial) example is the Division of Bads, where the UGE axiom places the (automatically satisfied) lower bound:

Si 2 un (ui; 1) = ui(w).

The Provision of a Public Good [problem (3)], on the other hand, has a ‘non-trivial’ positive group externality. When agent i alone utilizes the technology c to produce the public good, he reaches his free access utility level:

un (ui; 1) =max ui(u, c(u)). II20

(5)

When individual preferences are non-decreasing in the public good, the

‘However, in the context where money can be transferred across agents (so that someone may receive all the goods to distribute, but compensate the others out of his own pocket), the free access upper bound has bite: it even rules out the No Envy property. See the discussion in Moulin (1990a).

316 H. M&in, Unijoorm externalities

vector (un(u,; l),..., un(u,; 1)) is on or below the Pareto frontier.’ Thus, the UGE axiom yields the free access lower bound:

Si(U1,. . . , U,) 2_max ui(a, C(a)). II20

This axiom was first introduced by Foley in his classical essay [Foley (1967)]. Note that a similar lower bound can be defined for coalitions of agents, leading to the concept of a core.

There are many problems where the group externality is not positive and not negative either. Consider, for instance, the division of a vector consisting of one unit of good X, and one unit of bad Y. Then in the (unanimous) economy with two agents endowed with the following utility function:

we have

un(ui;2)=2.(1/2)-1/2=1/2<un(uj;1)=2.1-1.

Thus, at this profile the vector (un(u,; l), un(u,; 1)) is unfeasible (above the Pareto frontier). By contrast, at the profile

we have

un(ui;2)=-1/2>un(u;;l)=-1,

so that the vector of free access levels is below the Pareto frontier.

5. The cooperative production models

A lot of attention has recently been devoted to cooperative production models [see Cohen (1986) Mirrlees (1974) Moulin and Roemer (1989),

*This well-known fact is proven as follows. For each agent i denote by a, a solution of program (5):

un(u,; I) = ui(ui,c(a,)).

Let i* be such that a: is highest among the various a;. Consider the feasible allocation where a”

units of public good are produced and its cost fully covered by agent i* (others pay nothing). Its utility vector is Pareto superior (or equal) to the free access utility vector (un(u,;l) ,.._, un(u.;n)).


Roemer (1986)]. As far as our concepts of externalities are concerned, the key distinction is between the cases of increasing and decreasing returns to scale.

Recall that our n agents share a technology transforming x units of the (private) input into y= f(x) units of (private) output. Agent i’s preferences are convex and are represented by a utility, ui(xi,yi), decreasing in xi and increasing in yi. One computes easily:

0 the free access (utility) level:

un (ui; 1) =max u~(x, f(x)); X20

(6)

l the unanimity (utility) level:

un (Ui; U) = max Ui(X, f(n. X)/n).

X20

(7)

Note that (7) can be read as: agent i is given a (l/n)th share of the technology. The following simple facts are proven in Moulin (1989a). The cooperative

production problem has a positive preference externality iff f is concave (resp. has a negative preference externality iff f is convex). The problem has a negative group externality iff f is subadditive (f(x +x’) 5 f(x) + f(x’)), and has a positive group externality iff f is superadditive (f(x + x’) 2 f(x) + S(x’)). Remembering that for a function f such that f(O)=O, concavity implies subadditivity, whereas convexity implies superadditivity, we obtain two especially interesting cases of the joint production problem:

Case 1. Concave production function: Problem 5. Here we have positive preference externality and negative group externality.

The two axioms of uniform externalities yield the decentralized bounds:

Case 2. Convex production function: Problem 6, or the Natural Mono- poly Problem [Ramsey (1927) Sharkey (1982)]. Here we have a negative preference externality and a positive group externality:

max Ui(X, f(X)) 5 Si(Ul, . x

.,u,)~m~~i(x.:,(nx)).

318 H. Moth, Uniform externalities

See Moulin (1989a) for further discussion and a generalization of these results.

Several authors [Mirrlees (1974), Moulin and Roemer (1989), Roemer (1986)] have discussed a particularly simple version of this problem with a single input (labor), a single output (corn), and where the two agents differ only by their skills. To fix ideas, say that the High skill agent converts one hour of labor into two units of input, but the Low skill agent gets only one unit of input per hour of labor; their preferences over (leisure, corn) are identical. Denoting by u this common utility and by L each agent’s endowment of leisure, we derive the following (different) preferences for the two agents over (labor, corn):

%IkY)=u ( 1 L-;,Y ; u,(x,y)=u(L-X,Y).

Consider an arbitrary solution S. In the discussion of this problem, Roemer (1986) proposes the following axiom, which he calls insurance against negative externalities. Calling the High skill agent Able and the Low skill agent Infirm:

Able should not benefit from sharing the technology with Infirm rather than with another Able agent. On the other hand, Infirm should not suffer from sharing the technology with Able rather than with another Infirm agent.

This is formally written as:

Sn(u,; uL) 5 Sn(u,, un) = un (G; 2)

SL(an; uL) 2 &(u,, uJ = un (u,; 2),

[Notice that Si(ui,ui) is precisely agent i’s unanimity level].

(8)

(9)

Thus, Roemer’s axiom says that Able should enjoy a negative (non- positive) preference externality while Infirm should enjoy a positive (non- negative) externality. It is incompatible with our UPE axiom [more precisely, in the DRS case, UPE yields exactly the inequality opposite to (8); and in the IRS case, UPE yields the inequality opposite to (9)].

The point is that Roemer’s axiom gives an ethical orientation to skills: if I am less skilled I need to derive some benefit from your different skill, but if you are more skilled you are not entitled to any such benefit. By contrast, our Uniform Preference Externalities axiom sees only two agents with different marginal rates of substitution between the input and output and requires that none of them suffers from these differences in taste. So in the DRS case, High skill is entitled, like Low skill, to benefit from the difference in skills; and in the IRS case even Low skill accepts a welfare reduction because of the difference in skills.


t

SJULrUJ uL

Fig. 1

See fig. 1 where the two axioms, UPE and Roemer’s Protection of Low skill [namely inequalities (8) and (9)], are compared. The figure shows the feasibility frontier of a typical IRS problem. It also shows the interval on the Pareto frontier corresponding to the UPE axiom, and the interval bounded by PL and the ‘incentive’ constraint S,(u,, u,) zS,(u,, UJ [the High skill agent should not end up worse off than the Low skill agent - see Mirrlees (1974)]. Notice that the incentive constraint is automatically satisfied when the UPE axiom is (this is easily checked both in the DRS and in the IRS case). This strengthens the interpretation of UPE in this problem.

6. Compatibility of the two uniform externalities axioms

We have defined two simple tests of fairness for resource allocation, but are they always compatible?

Assume first that we are given a problem where the preference externality is always positive, whereas the group externality is always negative (like Division of Goods and DRS Production). In such a problem it must be


Table 1

Preference Group externality + - texternality

1 Division Division of + of Bads Goods,

DRS production

IRS Production Public Bad _ Public Good Provision

Provision

true’ that for any preference ui in D, un(ui; n)sun(ui; 1). Hence, the unanimity lower bound and the free access upper bound are compatible at every profile (that is, we can find a Pareto-optimal and anonymous mechanism meeting those two bounds).

A similar argument shows that in a problem with a negative preference externality and a positive group externality (such as Provision of Public Good, or IRS Production), the unanimity upper bound and free access lower bound are always compatible.

Consider next a problem where both the preference externality and the group externality are positive (like in Division of Bads). As noted at the end of section 3, the unanimity level is the highest decentralized lower bound function, hence it is above the free access level (as the latter is a decentralized lower bound as well). Therefore the UPE axiom implies the UGE axiom in those problems (in particular they always are compatible). In a problem where both the preference externality and the group externality are negative (like in Provision of a Public Bad), the same conclusion (UPE implies UGE) holds, because the unanimity level is then the lowest decentralized upper bound function.

Thus, when a problem has both its preference externality and group externality of a constant sign, the two uniform externality tests alwqs are compatible. By contrast, when the sign of either one of these two externalities varies, all conceivable configurations may happen: the UPE and UGE axioms may be incompatible, the comparison of un(u,;n) and un(ui; 1) may vary from one preference to another, and so on. Demonstrating examples can be cooked in the joint production model by playing with production technologies where returns to scale are neither increasing nor decreasing, see Moulin (1989a).

Table 1 summarizes the sign of our two externalities, in the six representative problems.

‘Consider the unanimity prolile (u,,. , uJ. The equal utility vector un(ui; 1) is below the Pareto frontier (by positive preference externality) and the equal utility vector un(u,; 1) is above (by negative group externality).


7. Comparison with no envy and resource monotonicity

No Envy was invented in the late sixties [Foley (1967)] and soon became the leading test of equity in ordinal allocation problems [see the survey paper by Thomson and Varian (1985)]. A most appealing feature of the concept (that is shared by our two uniform externalities axioms) is to rely on ordinal, non-intercomparable individual preferences.

Resource Monotonicity was introduced recently as a consequence of common ownership of the resources [Moulin (1987, 1988a), Moulin and Thomson (1988), Roemer (1986)]. When the resources grow (the set of feasible allocations expand) this axiom requires that the welfare of no agent should decrease.

We illustrate in our six problems the following claims: within the class of Pareto-optimal and anonymous mechanisms, the UPE axiom is compatible with No Envy but incompatible with Resource Monotonicity; the UGE axiom, on the other hand, is compatible with Resource Monotonicity but incompatible with No Envy.

For the first illustration of these themes we take the Provision of a Public Good [problem (3)]. Here No Envy of an allocation (a, cl,. . . , c,,) simply means uniform cost shares ci=c(a)/n (as every agent consumes the same amount of the public good). Under standard convexity assumptions, a Pareto-optimal and envy-free allocation exists [see Thomson and Varian (1985)]. Clearly, all envy-free allocations meet the unanimity upper bound [since agent i’s unanimity utility (3) is his favorite uniform cost shares allocation]. So No Envy implies the UPE axiom. On the other hand, a Pareto-optimal allocation with uniform cost shares often violates the free access lower bound (5). Think of a profile where an agent derives no utility whatsoever from the public good; he will have to pay (l/n)th of the cost of whatever quantity of public good it is efficient to produce, resulting in a net loss of utility.

The free .access lower bound (5) is compatible with Resource Monotonicity (meaning here that when the cost function decreases, the welfare of no agent decreases); in fact the combination of these two axioms uniquely characterizes a mechanism called egalitarian equivalent [Moulin (1987)]. This mechanism shares costs in such a way that every agent is indifferent between his current allocation and a certain quantity (the same for every agent) of free

public good. Finally, we note that Resource Monotonicity is incompatible with the unanimity upper bound (3). This is proven by an argument similar to that of the main theorem [Moulin (1987)].

We turn now to the Division of Goods [problem (l)]. An allocation of the resources o into (non-negative) shares z 1,. . . , z, is said to be envy-free if we have:

Ui(Zi) >= Ui(Zj), for all i, j = 1,. . . , n. (10)


An envy-free allocation is not necessarily Pareto optimal (for instance, equal split is envy-free) nor does it always satisfy the equal split lower bound. However, when preferences are convex (as we assume throughout, to ensure that our problem has a positive preference externality), an allocation can always be constructed that is envy-free, Pareto optimal and meets the equal split lower bound. It is the celebrated competitive equilibrium from equal split (CEES).

Suppose we convert the commonly owned resources o into n identical shares, w/n, of which each agent privately owns one. Any competitive equilibrium from this initial endowment satisfies the above three properties.” Those are the main arguments for his popular solution of the Division of Goods problem [see Dworkin (198 l), Maskin (1987), Thomson and Varian (1985)].

Note that the free access upper bound is vacuously true for every division mechanism (since everyone receives a non-negative share, everyone gets at most 0). So in this problem, No Envy is compatible with the free access bound. However, it can be shown that No Envy is not compatible with any of the minimal decentralized upper bound functions of the Division of Goods problem [see Moulin (1988b, 1989a)].

The Resource Monotonicity axiom is incompatible with Uniform Prefer- ence Externalities. It is shown in Moulin (1988b) that any (Pareto-optimal) mechanism guaranteeing an equal split utility to every agent must generate the growth paradox at some profiles (more cake to share but a smaller share for me): in particular it cannot be resource monotonic (more cake to share means more utility from my share). On the other hand, Resource Monotoni- city is compatible with any minimal decentralized upper bound function [see Moulin (1988b, 1989a)].

Our next example is the cooperative production of a private good [problems (5) and (6)]. No Envy is defined here by the same inequalities (10). Notice that envy bears on the net trades (input contribution, output shares), rather than the final consumption of input, output. Our interpretation is valid when input is a homogeneous good transferable across agents (so it is not appropriate to think of labor as the input). Our definition avoids the pitfalls of the classical definition involving comparisons of final input consumption [see Panzar and Schmeidler (1974)].

Assume that the function f is concave [problem (5): DRS Production]. Much like in the Division of Goods problem, we can construct a Pareto- optimal allocation that meets No Envy and the unanimity lower bound: it is a competitive equilibrium of the economy where each agent owns (l/n)th of the single firm.

“Conversely, in an economy with many small agents, the only envy-free and Pareto-optimal allocations are the CEES ones [Varian (1976)]. Thus, in this limit case, No Envy plus Pareto optimality do imply the equal split lower bound.


Fig. 2

On the other hand, the competitive equilibrium with equal shares of the firm may violate the free access upper bound. More generally, at some profiles there may not exist any Pareto optimal and envy-free allocation meeting the free access upper bound. The simplest example involves three

agents. The production function is

f(x) =min {x, 1).

Agent l’s utility is linear:

Preferences for agents 2 and 3 are depicted in fig. 2. Suppose, by way of contradiction, that the allocation (z,, z2, z3) is Pareto optimal, and meets No envy and free access upper bound. Assume first that x1 ~0. By Pareto optimality both z2 and z3 must be at a kink of the indifference curve of agents 2 and 3, respectively. On the other hand, the upper bound implies:

where z!j and zi are depicted in fig. 2. Thus, we must have x2 2 x!, xj 2 x:, and hence x2+x, > 1, and input is in excess, so x1 >O contradicts Pareto


optimality. Thus, xi =0 (agent 1 does not contribute any input), so zi =

(O>Y,). By construction, agent 3’s preferences have a kink at w =( 1,0.4). Hence,

agent l’s indifference curve through w intersects the vertical axis at (0,0.15). Check that we must have y, SO.1 5. Otherwise

contradicting No Envy. We now claim uj(zj)~u3(w). Otherwise z3 is somewhere on the dotted line

between w and zi, hence above the indifference curve of ui at (0,0.15), and therefore agent 1 envies agent 3. Summarizing we have:

~,(Z,)~~,(O,O.15), UZ(ZZ) Z%(8)> u&J Z%(W).

This in turn contradicts Pareto optimality since the following allocation is feasible:

z; = (0,0.2), z; = (0,0.4), z>=w=(1,0.4).

Finally, we discuss Resource Monotonicity. Combined with the free access upper bound, it characterizes a unique mechanism called ‘constant returns equivalent’ [Moulin (1990a, 199Oc)]. It can be shown that no (Pareto- optimal) mechanism satisfies both the Uniform Preference Externalities axiom and Resource Monotonicity [see Moulin (1990a)l.

8. Concluding comments

The unanimity utility (un (ui; n)) and free access utility (un (ui; 1)) are two decentralized functions: by this we mean that they depend on agent i’s preferences, not on other agents’ preferences. We have provided a variety of examples where either one or both of these functions can always be taken as a lower bound (or always as an upper bound) of any agent’s welfare. That is, the utility vector (un(u,;n),..., un (24,; n)) [resp. the utility vector

(un(n,;l),..., un(u,; l))] always lies on or above the Pareto frontier (or it always lies on or below it). This suggests a more general question.

For a given resource allocation problem, which decentralized functions +(uJ are possible lower bound (or upper bound) functions for some Pareto optimal (anonymous) mechanism? In other words, which functions $ (asso- ciating a utility level to each utility function in the preference domain) are such that at every possible profile (u,, . . . , UJ the utility ($(u,), . , $(u,)) is on or below the Pareto frontier of the corresponding economy? And which functions $ are such that ($(u,),.. ., $(u,)) is on or above the Pareto frontier


of any economy (u,, . . . , u,)? These two questions are addressed systematically in a companion paper [Moulin (1989a)]. In the Division of Good problem, Moulin (1988b) solves a similar question with a variable number of agents and a fixed per capita pie.

It turns out that the unanimity utility, un(u,;n), is a crucial component of the answer; in all problems with a positive preference externality it is the highest (decentralized) lower bound function; in all problems with a negative preference externality, it is the lowest upper bound function. The general approach also sheds some light on the free access utility function, un(u,; l), in the problems of cooperative production with multi inputs and multi outputs.

Another direction in which the methodology of the current paper can be extended is the variable population context. Recall that the UGE axiom compares an agent’s welfare in the actual society to his (virtual) welfare, should every other agent vanish. Think now of the other agents vanishing one at a time and look at the corresponding sequence of our (same) agent’s welfare. In the Provision of Public Good problem, we would expect to find a decreasing (at least non-increasing) sequence. In the Division of Goods, we would expect to find an increasing (non-decreasing) sequence. This kind of Population Monotonicity property was originally introduced by Thomson in the context of axiomatic bargaining [see Thomson (1983a, 1983b) and Chan (1986)]. In the resource allocation problems that we have discussed here, Population Monotonicity is a natural strengthening of the Uniform Group Externalities axiom. See Moulin (1990a) for a discussion in Division of Goods and DRS Production. See Moulin (1989b) for a discussion in Provision of a Public Good (or Bad). See Moulin (1990d) for a discussion in Cooperative Production. Finally, Moulin (1990b) and Sprumont (1989) discuss this axiom in the context of cooperative games with transferable utility.

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Uniform externalities: Two axioms for fair allocation