To simplify let’s assume that λ<1. Then the demand x1 belongs to (0,1) for r3 <
α<r1, with
(1 + λ)(1 + δλ) - \''^3 (1 + λ)(1 + δλ) - √δt1 ∕κθ∖
r3 = 2δ(λ + δλ + δ)2 , r1 = -----------2-----------, (52)
∆r1 = 1-4δ2λ+2λ(1-α)+λ2(1+δ2)+2λ3δ2(1+δ)+δ2λ4 (53)
∆r3 =(1+δ2λ2)(1 + λ2) + 2λ(1 - λ2 - λ(1 + δ2) + δ(1 + λ)) (54)
Moreover, r1 tends to 1 for δ → 1 while r3 can be larger than 1 for λ close to 1 and
δ<0.5.Whenα is smaller than r3, then the SPE demand is a corner solution x1 =1.
When α is larger than r1 (however, this case is not interesting for δ → 1), then we
have a corner solution of the system of indifference conditions and x1 =0. For λ<1,
the demand y2 defined in (51) is always positive, however, it will be also larger than
1 unless λ is close to 1 and α is sufficiently large for δ large.
The conditions becomes more transparent forδ→1,then the demands (50) and
(51) becomes
λ2 + 2λ - α
2λ(1 + α)
(55)
(56)
1 + 2λ - λ2α
2λ(1 + α)
the demand lx1 is in (0,1) for ʌ/! + α — 1 < λ < α + ʌ/a(l + α). Moreover, for
λ < 11 + α — 1, then lx1 < 0 while for λ > α + ʌ/a(! + α), then lx1 > 1. The
demand ly2 is in (0,1) for
-y6α(1+α) -α
α
<λ<
. Moreover, for λ < √α(1+α)-α,
αα
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