2.1 Type-exclusive assignment economy

Consider an economy that has a population of agents and a population of indivisible goods. Each indivisible good is labeled with a single type, g, which describes the different characteristics that fully characterize a good. The set of types of goods is G . Each agent is labeled with a type, a, where a particular type represents the preference relation ≾a over G . The set of types of agents is A . Assume that A and G are compact Borel spaces; that is, they are separable and compact metric spaces.

We assume that preference relations ≾aa∈A are represented by a continuous utility function u : A × G → [0, 1] , that is:

Two comments are in order about the utility function: 1) continuity is a requirement one cannot dispense of for our results to hold (the definition of continuity should be understood in the context of general topology), and 2) the utility function u (·, ·) represents the preference relations of all types of agents, so when one considers transformations f of u (·, ·) that also represent the preference relations of all types, they might treat arguments a and g differently. Specifically, since no cardinality of utility is assumed for our purpose, any transformation uf(·, ·) = f (u (·, ·)) that is strictly increasing also represents the original preference relations, that is, if g´≾ag´ , then:

This is the only monotonicity restriction required on f .²

Let B(A) and B(G) be the Borel σ-algebras of A and G , respectively. Probability measures η and ν assign a population distribution over the sets A and G , respectively. Finally, we denote the population of agents and the population of indivisible goods by the probability measure spaces:

and

respectively. A type-exclusive assignment for the economy E is a measurable function u : A → G. A type-exclusive assignment µ for E is TEEUU if for each set of types of indivisible goods E in B(G) , the amount ν(E) of indivisible goods is proportional to the amount η μ-1(E) of agents. In other words, a type-exclusive assignment μ for E is TEEUU if:

A type-exclusive assignment economy is a quadruple E := (A, G, u, μ0) , where A is a population of agents as in (3), G is a population of indivisible goods as in (4), u is a continuous function that satisfies (1), and, finally, μ0 : A → G ∪ ∅ is a measurable function which assigns for each type of agent a in A the agent’s initial endowment μ0 (a) in G and satisfies (5), or all agents have the empty set ∅ as initial endowment. The function , μ0 is called the initial type-exclusive endowment.

The following are two examples of type-exclusive economies and assignments.

Example 1: Consider an economy E where A and G are finite sets with the same cardinality n and the function u in (1) is represented as a square matrix [u (a, g)]a∈A,   g∈G of rank n . Let η and ν be uniform probability distributions over the sets A and G, respectively; n that is, η (a) = v (b) =1n for all a ∈ A and g ∈ G . In this case, any bijective function μ : An → G is a feasible type-exclusive assignment.

Remark: The economy in Example 1 is a particular case of ^{Shapley and Scarf (1974)}.

Example 2: Consider an economy E where A and G are both the interval [ 0, 1 ] and the function u in (1) is u (a, g) = a2+ g2 . Let η and ν be uniform probability distributions over the sets A and G , respectively; that is, η (da) = ν (dg) = 1 for all a ∈ A and g ∈ G , where η (da) and ν (dg) are density functions. In this case, the functions μ1(a) = a and μ2a=1-a are TEEUU type-exclusive assignments for E .

We define now the core for this economy.

Definition 3: The core C (E) of an economy E is the set of all TEEUU type-exclusive assignment µ such that there is no coalition S∈B(A) with (ν (S) > 0) and type-exclusive assignment γ that satisfies the following three conditions:

E1. i) η(γ-1(E)) = ν(E) for all E ∈ B (G) ∩ γ (S)- where γ (S)- is the topological closure of γ(S) ; ii) μ(S) = γ(S) , where μ(S)- and γ (S)- refer to the topological closures of μ(S) and γ(S) , respectively.

E2. u(a, μ(a)) ≤ u(a, γ(a)) η -almost everywhere in S.

E3. There exists D ∈ B (A) ∩ S with η(D) > 0 , and u (a, μ(a)) < u (a, γ (a)) η -almost everywhere in D .

Condition E1-i) refers to the feasibility of the assignment; a mass of goods is assigned to a mass of agents in equivalent proportions. Put differently, 30% of the agents cannot be assigned to 65% of goods. Condition E1-ii) ensures that the blocking coalition S does not require goods held by agents out of S. Conditions E2 and E3 refer to the incentives that individuals in coalition S have to improve with respect to their assignments. In conditions E2 and E3 the statement “η-almost everywhere” means that these conditions can fail only in a subset of η-measure zero.

2.2. Looking for a Pareto optimal assignment

In this section, we consider a social planner who searches for a typeexclusive assignment that is Pareto optimal (when the initial matching is the empty one, i.e., μ0 (a) = ∅ for all a in A ). Let L be the set of all feasible type-exclusive assignment, i.e.,

Consider the social planner’s problem in an economy E :

with L as in (6). We observe that L can be empty, as in Example 6.

If μ* is a solution to the planner’s problem, one cannot strictly increase the utility of a given type without lowering the others; thus μ* is Pareto optimal. Obviously, it is not the only one, and when u is transformed by a new utility function that satisfies (2), the new solution might be another Pareto optimal assignment. The dependence on the utility representation is not critical for our present purpose, which is to establish the relation between the core of the economy C (E) and the social planner’s problem (7).

Theorem 4: A type-exclusive assignment μ* is solution to the social planner’s problem (7), then μ* is in C (E) .

Proof. Suppose that μ* is solution to (7) and it is not in C (E) . Then there exists S ∈ B (A) (with η (s) > 0 ) and a feasible typeexclusive assignment γ which satisfy E1-E2 in Definition 3.

Now, consider the exclusive assignment:

Note that γ (S)-=μ (S)-=μ*S)- . Let E ∈ B (G) . Then by E1-ii), the properties of the inverse image, and (5), we have that:

Thus, μ is a TEEUU exclusive assignment for E . Moreover, by E2 and E3, it satisfies that:

Therefore, μ* is not optimal for (7), which is a contradiction. So, we conclude that μ* is in the core.

Example 5: Consider an economy E as in Example 2. In this case, L (as in (6)) is the set of all bijective functions μ : A → G . The social planner’s problem is given by the optimization problem:

2.3. The core and TEEUU allocations

Consider an economy E . If μ* is a solution to the problem (7), then by Theorem 4, μ* is in C (E) and therefore the core of E is not empty. Nevertheless, the set of TEEUU allocation L in (6) is not necessarily compact nor convex; in fact it may be empty (as in Example 6). In any case, (7) may have no solution. The following example provides a case where the set of TEEUU allocations L is empty.

Example 6: Consider a population of agents A := (A, B (A) , δa) and a population of goods G := (G, B (G) , ν) , where  δa is Dirac probability measure at a ∈ A and ν is defined by:

where δg1 and δg2 are Dirac probability measures on G with g1 /= g2 . This example describes a situation in which we only have two types of goods, and one type of agent. In this case L = ∅ .

Remark 7: Consider an economy as in Example 1, where A and G are finite sets with the same cardinality n , and η and ν are uniform probability distributions over the sets A and G , respectively. In this case, the social planner’s problem has a solution; see, for example, ^{Koopmans and Beckmann (1957)} or ^{Shapley and Scarf (1974)}. Moreover, according to Theorem 4, the core of this economy is not empty.

2.4. Non-atomic sets of types and the non-emptiness of C ( E )

In this section, we establish particular conditions under which if we have non-atomic sets of types, then the core of an economy E is nonempty. We consider the following assumptions:

A.1. Non-atomic sets of types. The set of types of agents A and the set of types of indivisible goods G are compact subsets of Rn , and η is a probability measure on B(A) , which is absolutely continuous with respect to n-dimensional Lebesgue measure.

A.2. Heterogeneity on utility. Let U be a differentiable function in A × G . If g', g ∈ supp (ν) with g≠g´ , where supp (ν) denotes the support of probability measure ν , then:

A.3.Convexity/concavity in types of agents. The set A is convex, and for each g ∈ supp (ν) , the function a → u (a, g) is strictly concave or strictly convex.

A.4. Boundedness on the heterogeneity of types of agents. The set int(supp( η )) is not empty and its complement is Lebesgue negligible;³ for every g ∈supp(v), a → (a, g) is differentiable and for any a ∈ supp(η) , there exists a neighborhood V of a and a number ca> 0 , such that:

Proposition 8: Consider assumptions A1, A2, and A3. Then the problem (7) admits at least one solution.

Proof. See ^{Levin (2004)}, Theorems 1.2 and 1.3.

A few comments on the conditions are in order. A1 Non-atomic set of types: previous results do not require the assumption; however, it might be imposed from the outset of our model since most economic models deal with continuousand even differentiabledistribution functions of types. Without A1, building a function that copes with TEEUU is technically difficult. A2/A3 Heterogeneity on utilities and heterogeneity in the types of agents; in both cases, a lack of heterogeneity leads to the same evaluation of two types of agents/goods by the objective function in the Pareto optimality problem (7), thus leading to the assignment of the same types in the solution, violating TEEUU. A3 can indeed be relaxed. Nevertheless, the changes in heterogeneity in types of agents should not be too steep, as imposed in A4.

Proposition 9: Consider assumptions A1, A2, and A4. Then the problem (7) admits at least one solution.

Proof. See ^{Levin (2004)}, Theorem 1.4. ^{Carlier (2003)} proposes similar conditions of Proposition 9 for metric spaces.

The next example satisfies the assumptions A1, A2, and A4. The reader can find other interesting examples in ^{Levin (2004)}.

Example 10: Let E := (A, G, u, μ0) be a type-exclusive assignment economy, where A and G are convex and compact subsets of Nn; η is absolutely continuous with respect to the Lebesgue measure on A; u (a, g) = -∑(ai - gi)2 for a := (a1, . . . , a2) and g := (g1, . . . , gn) , and µ0 is any agent’s initial endowment. Let μ* (a) = Ha + b , where H is symmetric and positive semi-definite matrix, and b ∈ Rn . If ν (E) = η μ-1(E) for all E ∈ B (G) , then μ* is the unique optimal solution of (7), and it is in the core of E .