Markets with Multidimensional Private Information

Impose Assumption 1 and assume there exists a β ∈ B with βΓ(v) > v. The semi-separating equilibrium is characterized by a discount factor for the marginal buyer, β̂ ∈ B, which is determined in equation (A1) below. For now, fix β̂ and assume, as we verify below, that β̂Γ(v) > v. We next define two critical prices. The lowest price with trade is p ≡ β̂Γ(v), the value that the marginal buyer places on an asset sold by the seller with the lowest continuation value. Note that, given our assumption, p > v, so a seller with the lowest continuation value strictly prefers selling his asset for p rather than retaining it. The second critical price is the highest one with trade. Let p̄ be the smallest price satisfying p̄ = β̂Γ(p̄), or p̄ = ∞ if there is no such price. That is, β̂Γ(v) > v whenever v < p̄. In the semi-separating equilibrium, the equilibrium buyer-seller ratio satisfies