By Stephen Morris
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Extra resources for Econometrica 2009 Vol.77 N° 1
If the indifferent type q for the lower offer p were below β, the next offer would at most be accepted by all types up to β, and type q would eventually accept an offer not exceeding p. But, provided p is close enough to p, type q would then strictly prefer to accept the lower offer p immediately. This shows that the indifferent type for p is indeed β. The last insight in the proof is that this scenario can be repeated for q ∈ (β2 β]. Let us elaborate on this. A consequence of the previous argument is that an offer of c(β) is accepted by all types up to β (and all offers after that are losing).
If buyer n makes an offer p < c(ql ) with independent type q, one has q ∈ (ql+1 ql ); thus the offer p is followed by offers equal to c(ql ). 18 If instead buyer n makes an offer p > c(ql ), say c(ql ) < p ≤ c(ql−1 ), the situation is a bit more subtle. We know that the indifferent type q associated with the offer p is at least equal to ql (for otherwise, all subsequent offers would be equal to c(ql ), and type q would strictly prefer to accept the current offer). If q > ql , then the offer p is followed by offers equal to c(ql−1 ), which type q accepts; hence q is given by p − c(q) = δ(c(ql−1 ) − c(q)) This equation has a solution in (ql ql−1 ] if and only if (1 − δ)c(ql ) + δc(ql−1 ) < p ≤ c(ql−1 ) 18 If it exists.
This is because the traditional point of view emphasizes the inability of the buyer to commit to not making another offer. Instead, the driving force here is the inability of the seller not to solicit another offer. This leads to a collapse in the probability of trade and an increase in the inefficiency. 2. Sketch of the Proof We here provide a sketch of why all equilibria result in an impasse. For simplicity, here we let the cost and the valuation functions be given by c(q) = q PUBLIC VS. PRIVATE OFFERS 39 and v(q) = (1 + α)q, where α ∈ (0 1).