Assortative Matching and Nontransferabilities: Beauty and the Beast

Appendix Equilibrium The economies we study have a continuum of individuals who are designated by the sets I = [0, 1] and J = [2, 3] with Lebesgue measure λ. The set I is the set of principals and the set J is the set of agents. The description of a specific economy includes an assignment of individuals to types via maps π : I −→ P, and α : J −→ A, where P and A are compact real intervals. The maps π and α are simple measurable functions. A matching correspondence is a function m : I −→ J satisfying measure consistency: for any measurable subset S of I, λ (S) = λ (m (S)) , that is the measure of principals is equal to the measure of their partners. Definition 1. An equilibrium specifies a matching correspondence m and payoff allocations u∗ : I −→ R+ and v ∗ : J −→ R+ that are measurable and satisfy the two following conditions. (i) Feasibility of (u∗ , v ∗ ) with respect to m : for almost all i ∈ I, u∗ (i) ≤ φ (π (i) , α (m(i)) , v∗ (m(i))) . (ii) Stability of m with respect to (u∗ , v ∗ ) : there do not exist (i, j) ∈ I × J and payoffs (u, v) such that u ≤ φ (π (i) , α (j) , v) , with u > u∗ (i) and v > v ∗ (j) . Because φ (p, a, v) is strictly decreasing on v ∈ (0, ψ (a, p, 0)) , an equilibrium satisfies equal treatment. Lemma 2. Let (m, u∗ , v ∗ ) be an equilibrium, then there is equal treatment: ¡ ¢ ¡ ¢ ∀i, i0 ∈ I, π (i) = π i0 ⇒ u∗ (i) = u∗ i0 ¡ ¢ ¡ ¢ ∀j, j 0 ∈ J, α (j) = α j 0 ⇒ v∗ (j) = v∗ i0 Proof. Suppose that there are two agents j and j 0 of type a getting different equilibrium utilities v ∗ (j) > v ∗ (j 0 ), and that j’s partner i is of type p. Then i gets φ(p, a, v ∗ (j)) < φ(p, a, v ∗ (j 0 )), where the inequality follows from the fact that φ is strictly decreasing in v. By continuity, there exists > 0 such that φ(p, a, v ∗ (j 0 ) + ) > φ(p, a, v ∗ (j)); {i, j 0 } can therefore block the equilibrium, a contradiction. The matching correspondence M in the text can be constructed from m as M (p) = {α (j) , j ∈ J : ∃i ∈ I, π (i) = p and j = m (i)} . Because we have equal treatment, the equilibrium payoff functions u∗ and v ∗ depend only on the type of the individual, and it is enough to define type-dependent payoff functions u : P → R+ and v : A → R+ as u (p) = u∗ (π (i)) when π (i) = p for i ∈ I and v (a) = v ∗ (α (j)) when α (j) = a for j ∈ J. In addition to our assumptions we need to impose for existence that φ is strongly comprehensive. Definition 2. φ is strongly comprehensive if there exists b > 0 such that whenever 0 < u ≤ φ (p, a, v) , u − c ≤ φ (p, a, v + bc) for any c ∈ (0, u) . This condition is satisfied if, for instance, the marginal utility of consumption is bounded at autarky. That equilibria exist is then a simple consequence of the existence result in Kaneko and Wooders (1996). Proof of Proposition 1 We prove (1), the proof of (2) is similar.12 Suppose that GID holds and that there exists an equilibrium hM, ui such that M does not satisfy PAM. Hence, there exist p > p0 , p, p0 ∈ P and a > a0 , a, a0 ∈ A such that a0 ∈ M (p) and a ∈ M (p0 ) . Stability requires that neither hp, ai nor hp0 , a0 i can profitably deviate. Hence, we need ¡ ¡ ¢¢ ψ (a, p, u (p)) ≤ ψ a, p0 , u p0 ¡ ¡ ¢¢ ¢ ¡ ≤ ψ a0 , p, u (p) , & ψ a0 , p0 , u p0 (9.1) (9.2) Applying φ (p, a0 , ·) to both sides of the second inequality we obtain ¡ ¡ ¢¢¢ ¡ ≥ u (p) . φ p, a0 , ψ a0 , p0 , u p0 (9.3) since by feasibility of the equilbirium u (p) ≤ φ (p, a0 , 0). Apply now φ (p, a, ·) to both sides of the (9.1) ;by definition of the inverse map we have13 ¡ ¡ ¡ ¢¢¢ if u (p) ≤ φ (p, a, 0) u (p) ≥ φ p, a, ψ a, p0 , u p0 ¡ ¡ 0 ¡ 0 ¢¢¢ if u (p) > φ (p, a, 0) φ (p, a, 0) > φ p, a, ψ a, p , u p (9.4) and therefore combining (9.3) and (9.4) we obtain ¡ ¡ ¡ ¡ ¢¢¢ ¡ ¢¢¢ ¡ ≥ u (p) ≥ φ p, a, ψ a, p0 , u p0 . φ p, a0 , ψ a0 , p0 , u p0 But (GID) requires that φ (p, a0 , ψ (a0 , p0 , u (p0 ))) ≤ φ (p, a, ψ (a, p0 , u (p0 ))) , therefore 12 ¡ ¡ ¡ ¡ ¢¢¢ ¡ ¢¢¢ ¡ = φ p, a, ψ a, p0 , u p0 , u (p) = φ p0 , a, ψ a0 , p0 , u p0 For one sided matching problems, the proof of sufficiency of GDD for NAM is slightly more involved than for sufficiency of GID for PAM. See our working paper Legros and Newman (2003). 13 By feasibility u (p) ≤ φ (p, a0 , 0) ; however since we do not assume that φ is increasing in its second argument, we do not know whether u (p) is greater or lower than φ (p, a, 0) . and therefore u (a) = ψ (a, p, u (p)) , u (p) = φ (p, a, u (a)) , u (a0 ) = ψ (p0 , a0 , u (a0 )) and u (p0 ) = φ (p0 , a0 , u (a0 )) . That is the matches hp, ai and hp0 , a0 i are stable when payoffs are u. If in an equilibrium hM,ui matches hp, a0 i and hp0 , ai form, GID implies that the equilibrium payoffs (u (p) , u (a)) are on the Pareto frontier for coalition hp, ai and payoffs (u (p0 ) , u (a0 )) are on the Pareto frontier for coalition hp0 , a0 i . We have showed that under (GID), if in equilibrium there are two matches that violate PAM, it is possible to reassign partners in order to obtain PAM without modifying payoffs and therefore without violating stability. This falls short however of showing that starting from an equilibrium in which M does not satisfy PAM we can find a payoff equivalent equilibrium satisfying PAM. The next part of the proof shows a simple algorithm to construct a new equilibrium hM0 , ui where M0 satisfies PAM. It is easier to develop the argument in reference to agents rather than types. Let I and J be respectively the sets of agents of types in P and in A. To simplify the exposition, assume that the sets of agents are finite and let N and M be the cardinalities of I and J. Then, we can write I = {ik , k ∈ {1, 2, . . . , n}} and J = {jk , k ∈ {1, 2, . . . , m}} . Higher indexes correspond to lower values of the characteristic, that is agent ik has type p and agent il has type p0 where p ≥ p0 if and only if k ≤ l. We show that under GID, any equilibrium hM, ui is payoff equivalent to the equilib- rium where type payoffs are given by u and where the matching is such that for agents, the match of ik ∈ I is jk ∈ J, whenever k ≤ min (n, m) . Suppose that in the matching of the equilibrium hM, ui there exists an agent ik ∈ I who is matched with jl ∈ J, where l = 6 k. Let k be the first time this situation arises. Hence for all k 0 < k, the match of ik0 ∈ I is jk0 ∈ J. Let jl be the match of ik and ir be the match of jk ; by construction if ik has type p and jk has type a, jl has type a0 ≤ a and ir has type p0 ≤ p. By our previous claim, we can match ik with jk and ir with jl , keep the same payoffs for the four agents without violating feasibility or stability. Proceeding in this fashion we can reassign the agents in such a way that ik is matched to jk for all k ≤ min (n, m) . Call µ : I → J this matching : µ (ik) = jk if k ≤ min (n, m) , µ (ik ) = ∅ if k > min (n, m) . Define a subset I 0 of I to be connected if ik ∈ I 0 and ik+l ∈ I 0 imply that it ∈ I 0 for all k ≤ t ≤ k + l. If I0 and I1 are subsets of I, write I0 < I1 if for all ik ∈ I0 and il ∈ I1 , l > k. Similar definitions apply to subsets of J. By construction, to the set of types P corresponds a partition [I (p) , p ∈ P ] of I such that for each p, I (p) is connected, similarly to the set of types A corresponds a partition [J (a) , a ∈ A] of J where each element is connected. Consider p > p0 , by our construction, I (p) < I (p0 ) and by definition of µ, µ (I (p)) < µ (I (p0 )) ; that is agents in µ (I (p)) do not have lower type than agents in µ (I (p0 )) . It follows that the matching correspondence on types induced by µ satisfies PAM. This proves sufficiency of GID for PAM. Proof of Proposition 8 Note that condition (7.2) requires strict preferences. Let p > p0 , a > a0 . Matches (p, a0 ),(p0 , a) are not stable under (7.2): if h (p, a) > h (p, a0 ) and f (a, p) > f (a, p0 ) , p and a should form a match; if h (p0 , a0 ) > h (p0 , a) and f (a0 , p0 ) < f (a0 , p) , p0 and a0 sould form a match instead. This shows that all equilibria satisfy PAM. It is not possible to replace the strict inequalities by weak inequalities. For instance, suppose that h (p, a) is strictly increasing in a for all values of p but that f (a, p) is constant for all values of p and a. Then any matching is an equilibrium since agents are indifferent between all partners. However, equilibria are not payoff equivalent to PAM.