Appendix. The solution begins with Eq. (2.15) from the text, which we repeat here for 1, (A.1)

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1 Aenix Aenix A: The equaion o he sock rice. The soluion egins wih Eq..5 rom he ex, which we reea here or convenience as Eq.A.: [ [ E E X, A. c α where X u ε, α γ, an c α y AR. Take execaions o Eq. A. as o ime, we oain: E E E E [ X. A. This exression is a ierence equaion in he execaion erms. Use o he lag oeraor L allows he susiue: E LE, E L E P. A. We hereore oain: L [ L E P 0 A.4 Thereore, eiher we require ha E P 0 or we require ha he quaraic exression in L e ienically equal o zero. As a resul, a nonrivial soluion requires ha: L [ L 0. A.5 Denoe he roos o his quaraic equaion y, i,, hen: i

2 an >. A.6 From Eq. A.6, we can assure ha one roo is greaer han an he oher roo is less han. Furhermore, he general soluion o Eq. A. is given y: j j E P j c, A.7 wih c an eermine y an iniial-value coniion. In he secial case o E P P, we can hereore susiue j ino Eq. A.7 an oain: P c. A.8 The consan c or corresoning o he greaer roo is assume o e equal o zero o ensure a convergen sysem. Now eine <, where is selece as he smaller o he wo roos. In his case, is suose o e zero so ha Eq. A.8 is exress as: E P P c. A.9 We can re-exress i as: c P. Thereore, he raional rice execaion rocess is given y: E P P, A.0 j j an so See Sargen 987, Ch.9. In calculaion, we ao he ackwar meho o oain 0 < < an >, an hen selec o ensure a convergen sysem. However, i he orwar meho is aoe, we may oain anoher wo roos: 0 < < an >, an hen choose o ensure a convergen 0 < <, raher han he value o sysem. In our ollowing analysis, we only use he roery o. Thereore, eiher ackwar soluion or orwar soluion will lea o he same conclusions.

3 E P P an EP P. A. Susiue Eq. A. ino Eq. A., we oain: X. A. Aenix A: The ynamic equaion o he sock rice. To oain he sock rice s ynamic equaion, we irs have o comue he long-run equilirium. In he long run, all unexece shocks will no exis anymore an he sock rice in each erio will e equal o he long-run equilirium sock rice. Juging rom he aove, we hereore se u ε 0, an relace an y in Eq.A. o oain: αy AR c, A. where c α y AR. In aiion, we can rewrie he RE equaion as ollows:. A.4 Susiue Eq.A. an Eq.A.4 ino Eq.A., we oain: c [ u α ε α q ε. A.5 u

4 Aenix A: Derivaion o. From Eq.. an Eq..4, we have:, RE A.6. SB A.7 θ Diereniaing he aove wo exressions wih resec o,,, an hen exressing hem in marix noaion, we oain:, { } A.8 where enoes he SB equaion: θ { α u ε, A.9 enoes he arial erivaive o wih resec o : α θ [ u ε 4, A.0 5 an enoes he arial erivaive o wih resec o :

5 α θ[ u ε 4 α [ u ε 4. A. Thus, alicaion o Cramer s rule yiels erivaion o:. A. Aenix A4: Deerminaion o uner ieren yes o shocks. The issuing shock Susiuing ε 0 ino an, we have: 0, an > 0. A. θ u Thereore, > 0. A.4 θ u The ivien shock Susiuing u 0 ino an, we have: 0, an > 0. A.5 α θ ε Thereore,

6 θ α ε > 0. A.6 The margin-rae shock Susiuing u ε 0 ino an, we have: 4 < 0, an 0 <. A.7 θ θ Thereore, 4 θ < 0. A.8 Aenix A5: Deerminaion o h uner ieren yes o shocks. The issuing shock From Eq..7 an Eq..8, we have: u h. A.9 θ θ Thus, iereniaing h wih resec o, we oain: h u [ θ u θ [ { u u { u θ u }

7 } u > 0. A.0 u The ivien shock From Eq.. an Eq.., we have: h α ε θ θ. A. Thus, iereniaing h wih resec o, we oain: h αε [ θ αε θ [ { αε α { ε α ε α θ ε } ε < 0. A. α ε } The margin-rae shock From Eq..6 an Eq.., we have: h θ θ. A. Similarly, le us iereniaing he aove exression wih resec o, we oain: h θ var [

8 } [ { θ [ { θ } 0 < θ, A.4 since 0 < in his case.