payoff player i obtains, vi , is as follows, with i =1, 2.
v1 = δt1(λ1x + δ1nα1(1 - y))
(1)
v2 = δt2 (1 - x + δ2nα2λ2y) (2)
Our technical assumption is as follows. Let λ2 ≤ λ2 ≤ λ2 where
ʌ _ α2(1 - δ1)(1 - δ2) _ δ2(1 - δ1δ2)
λ2 = δ2(i - δ1δ2) an 2 = α2(i - δι)(i - δ2)
This assumption allows us to simplify the presentation. This is not a restrictive
assumption since in the most interesting case in which some frictions tend to disappear
(i.e., ∆ → 0), these bounds tends to include the entire positive real range (i.e., λ2 → 0
and λ2 → ∞).
2.1 Equilibrium
Let
_ λι(1 — δ1δ2 + α2λ2(i + δ2)(l — δι)) b _ λι[(1 — δ2)α2λ2 — δ2(l — δ1δ2)]
a = δ2(i — δ2) , = δ2(i + δι)(i — δ2)
P _ λι[(i — δ2)α2λ2δι — (i — δ1δ2)] _ λ1δ1(i — δ1δ2 + α2λ2(i + δ2)(i — δι))
f = (i+δι)(i — δ2) ’ g = (i—δ2)
For α1 that varies between the boundaries2 f ≤ b ≤ g ≤ a, we can define different
SPE demands. There are at most three SPE with immediate agreement (when 0 <
2The assumption λ2 ≤ λ2 implies that b ≤ g. When the frictions within the bargaining stage
tends to disappear, ∆ → 0,f= b and g = a.