Munich Personal RePEc Archive

Macroeconomic Implications of Modeling

the Internal Revenue Code in a

Heterogeneous-Agent Framework

Moore, Rachel and Pecoraro, Brandon

Joint Committee on Taxation

8 June 2018

Online at https://mpra.ub.uni-muenchen.de/91319/

MPRA Paper No. 91319, posted 07 Jan 2019 18:28 UTC

r♦♦♥♦♠ ♠♣t♦♥s ♦ ♦♥ t ♥tr♥

♥ ♦ ♥ tr♦♥♦s♥t r♠♦r∗

♦♦r† r♥♦♥ P♦rr♦†

♠r

strt

① ♣♦② ♥②ss ♥ tr♦♥♦s♥t ♠♦s t②♣② ♥♦s t s ♦

s♠♦♦t t① ♥t♦♥s t♦ ♣♣r♦①♠t ♦♠♣① ♣rs♥t t① ♥ ♣r♦♣♦s r

♦r♠s ♥ ts ♣♣r ①♣♦r t ①t♥t t♦ t t① t ♦♠tt ♥r

ts ♦♥♥t♦♥ ♣♣r♦ s ♠r♦♦♥♦♠ ♠♣t♦♥s r♥t ♦r ♣♦② ♥

②ss ♦ ♦ ts ♦♣ ♥ tr♥t ♣♣r♦ ② ♠♥ ♥ ♥tr♥ t①

t♦r ♥t♦ rs ♦r♣♣♥ ♥rt♦♥s ♠♦ tt ♦♥t♦♥♥

♦♥ ♦s②♥rt ♦s♦ rtrsts ①♣t② ♠♦s ② ♣r♦s♦♥s ♥ t

♥tr♥ ♥ ♦ ♣♣ t♦ ♦r ♥♦♠ ❲ ♥ tt ♦r t♦♥st♥t

st② stt ♥②ss ♦ ♥ t① ♣♦② ♥ ♦t ♣♣r♦s ♥rt s♠r

♣♦②♥ ♣ttr♥s ♦ ♠r♦♦♥♦♠ tt② s♣t rt♦♥ ♥ t ♥r

②♥ ♣ttr♥s ♦ ♦s♦ t①♣rrr ♦♥s♠♣t♦♥ ♥ ♦r s♣♣② ♦r

♦r ts rt♦♥ ♥ ♥r②♥ ♦r s ss♦t t s♥♥t q♥t

tt ♥ qtt r♥s ♥ ♠r♦♦♥♦♠ rts ♦♥ t♥♥

tr♥st♦♥ ♣t ♠♠t② ♦♦♥ ♣♦② ♥ ♦♥sq♥t② t♦

t s ♦ ♥♦♥t♦♥ s♠♦♦t t① ♥t♦♥s ♠② rs♦♥ ♠♦♥ s♠

♣t♦♥ ♦r st② stt ♥②ss ♦ t① ♣♦② t♦♥ s♦ t♥ ♦r tr

s ♥ tr♥st♦♥ ♣t ♥②ss t♥ tr♦♥♦s♥t ♠♦s

♦s ②♦rs ②♥♠ s♦r♥ t① ♥t♦♥s ♥ t♦rs tr♦♥♦s ♥ts

∗s rsr ♠♦s ♦r ♥rt♥ ♦r t st ♦ t ♦♥t ♦♠♠tt ♦♥ ①t♦♥ t s♠♠rs ♦ ♦t ♣rts ♥ ♦t ♦ss ♦ ♦♥rss ♦♠♣rs t ♦♥t ♦♠♠tt ♦♥ ①t♦♥ ts♦r s♦ ♥♦t ♦♥str t♦ r♣rs♥t t ♣♦st♦♥ ♦ ♥② ♠♠r ♦ t ♦♠♠tt s ♦rs ♥tr t♦ t ♦♥t ♦♠♠tt ♦♥ ①t♦♥ sts ♦r ♥ ts t② t♦ ♠♦ ♥ st♠t t♠r♦♦♥♦♠ ts ♦ t① ♣♦② ♥s

†❯ ♦♥rss ♦♥t ♦♠♠tt ♦♥ ①t♦♥ ♦r ♥ ❲s♥t♦♥ ♦♦rt♦ r♥♦♥P♦rr♦t♦ ♦rrs♣♦♥♥ t♦r

♥tr♦t♦♥

① ♣♦② ♥②ss ♥ ♠r♦♦♥♦♠ ♠♦s rqrs t ♥♦r♣♦rt♦♥ ♦ t① s②st♠tt ♥ ♣♣r♦①♠t ♦♠♣① ♣rs♥t t① ♥ ♣r♦♣♦s t♦♥s ♥ ♥s tt ♣♦② ♥s t t①♣②r r♥t② s♣t ♦ t ♦s♦ tr♦♥t② tr ♥ r② tr♦♥♦s♥t ♠♦s t ♥tr t① s②st♠ s ♦t♥♣♣r♦①♠t s♥ s♥ ♣r♠tr③ t① ♥t♦♥ tt s♠♦♦t② ♠♣s ♦s♦♥♦♠ ♥t♦ t① ts ♦r t t① rts s s t♦s ♦♣ ♥ str② ♥♦ ♦ ♥ trss é♥♦ ♥ ♥ rt

❲ t r♥t ♦r ♦ r t s ♣♦♥ ts trtr ② s♥ r ♥♠r ♦ t① ♥t♦♥s ♦♥t♦♥♥ ♦♥ ♦s♦ t♦ s♦rt♦♠♥s ♦ ts♦♥♥t♦♥ ♣♣r♦ r♠♥ rst ♥t♦♥ ♦r♠ ss♠♣t♦♥s ♥ t ♠♣♦st♦♥ ♦s♠♦♦t♥ss ♦♥ t t① s②st♠ r qst♦♥ ♣♣r♦①♠t♦♥s ♦ ♥ t t① s②st♠tt trs ♥♦♥♦♥①ts ♦♥ ♥ t♦ ①♣t② ♦♥t♦♥ ♦♥ ♦s②♥rt♦s♦ rtrsts s s ♥ stts ♥♠r ♦ ♣♥♥ts ♥ t①♣rrr♦♥s♠♣t♦♥ ♦s ♥♦rs t r♥ ♥ t① t② ♦r ♦s♦s r♥♥ s♠r♥♦♠s r♦♥ ♣ ♥trst ♥ t ♠r♦♦♥♦♠ ts ♦ t① ♣♦② ♥s ♦rt ❯♥t tts rr♥ts rt t♦♥ ♦ ts s♦rt♦♠♥s

♦ ①♠♥ t ①t♥t t♦ t s♦rt♦♠♥s ♦ t ♦♥♥t♦♥ ♣♣r♦ ♠♣t♦♥s r♥t ♦r t① ♣♦② ♥②ss ♦♣ ♥ tr♥t ♣♣r♦ ❲t♥ rs ♦r♣♣♥ ♥rt♦♥s ♠♦ ♠ t① t♦r tt ①♣t② ♥♦r♣♦rts t① ♣r♦s♦♥s ♦ t ❯♥t tts ♥tr♥ ♥ ♦ ♣♣ t♦ ♦r ♥♦♠ ♥ ♦♥t♦♥s ♦♥ ♦s②♥rt ♦s♦ rtrsts ♥♦♠♣t♥ t① ts ♣② ♠♦ t ♣rs♥t sttt♦r② t① rt s st♥r t♦♥ r♥ ♥♦♠ rt t① rt ♦♠ ♠♦rt ♥trstt♦♥ stt ♥ ♦ ♥♦♠ ss ♥ ♣r♦♣rt② t① t♦♥s rt ♥t♦♥ ♥t ♥st♠♥t ♥♦♠ ♥ r srt①s ♥ ♣♥♥t r rt❯♥ t ♦♥♥t♦♥ ♣♣r♦ ♦ ♥♦t ♠♣♦s ♥t♦♥ ♦r♠ ss♠♣t♦♥s ♦♥ tt t① rt ss ♥st ♥♦♥♦s ♦s♦ ♦r ♥rts t♦♥sr♦♠ t sttt♦r② t① rt s tr ② ♠♣♦②♠♥t ♦s ♦ tt① rts ♦r ♦♥s♠♣t♦♥ ♦s ♦ t t♦♥s s s ♠♣♦rt♥ts rt♥ ♣♦② ♣r♦♣♦ss ♠② ♥ ♠srs t ♣♦t♥t② ♦stt♥ ts♦♥ ♥♥ts r② r♦ss ♦s♦s r ♥ r♣s ♦r ♠② ♥trt♦♥s t ♥r②♥ ♣r♦s♦♥s ♥ t t① ♦ ② ①♣t② ♠♦♥ r♥ ♦ ② t① ♣r♦s♦♥s ♥ ♦ ♣rs♠♦♥② ♦r ♣♣r♦ ♥srs tt t① ♣♦②♥s ♥♦♥ r srt ts ♦♥ rt② s♠ r♦♣ ♦ ♦s♦s r ♥♦ts ♦t s s♠r ♥ ♦r t r ♣♦♣t♦♥

❲ ♦♦s ♦♥ ♦ t ♦♠♠♦♥②s s♠♦♦t t① ♥t♦♥s ♥②③ ♥ ♥r t

♦♥s♠♦♦t t① ♥t♦♥s ♥ s ♥ ❱♥tr ♥ t ♥ rstr♦♠ ♦r ①♠♣ ♥ ♣r♦s ♥ ♦ ♠ss♣t♦♥ ♦r t ♦ ♥ trss

t① ♥t♦♥♥ t ❯♥t tts ♦s ♦ ♣rs♥tts ♦♣t ❵②♥♠ s♦r♥ r ❳

s ♥♦r♣♦rt ♥t♦ ♦♥t ♦♥rr♥t t s♦t♦♥ ♦r t 114th ♦♥rss ♥ rr♠♥ t ♦s ♦r t 115th ♦♥rss s r rqrs ♣♦♥t st♠t ♦ t r♥ t ♦r rt♥♣r♦♣♦s st♦♥ tt ♥♦r♣♦rts t rs♣♦♥s ♦ ♠r♦♦♥♦♠ tt② ♦r ♠r♦♦♥♦♠ ♥②ss ♦ t r♥t② ♥t P

r s ♦ ❵♣rs♥t rrs t♦ t① ♥s t♦s ♥s t♦ t r♦♠♣ss ♦ P ♦♦q② ♥♦♥ s t ❵① ts ♥ ♦s t ♦

♦r ♥ r② sss♦♥ ♦♥ t s ♣♣r♦♣rt s ♦ sttt♦r② ♥ t ♠r♥ t① rts srr♦ ♥ s ♥ str② ♥ ♦

t é♥♦ t① ♥t♦♥ s ♥♠r t t♦ ts ♦r ♣♣r♦♦rs t♦ ♥♥♦t♦♥s rst t ♥tr♥ t① t♦r ♥ trt s♣ ♦s♦st ② ♣♦② ♥ ♦♠♦♥rs♣ s ♦ ♥ t ♠♦ s♦ t ♦♠♠♦rt ♥trst t♦♥ r r♣ ♦r ts r t♦ r t♦ ♥trt♦♥s t ♥♦tr ♥ t♦ t t① ♦ ♦r ①♠♣ ♦♥② ♦s♦s ♠♥tt t♦♥ ♦ ①♣r♥ ♥ ♥rs ♥ tr t① t② tr♥t② t t①♥t♦♥ ♦ st ♦r r② ♦s♦ ♥sr♠♥t② s♦♥ ♥t ♦ tt① t♦r s tt ♦s♦s t ♦♣♣♦rt♥t② t♦ rt ♦♣t♠② t♦ t ♣♦②♥ t t♦♥ s r♣ ♦r ts s r rt② ♦r ♥rt② r♥trs♦ ♦ ♣rs ♦s ♥ t ♥r tr ♥r ♣rs♥t ♠② ♦♥t♥ t♦ r♥t♦r t♦ ♣rs s♠r ♣r♦♣rt② ❯♥r t t① ♥t♦♥ t ①t♥t t♦ ♦s♦s r t ② t ♥ s ♠♣t② ss♠

❲ s♠t t♦ t① ♣♦② ♥s s♥ t ♥tr♥ t① t♦r ♥ t s♠♦♦tt① ♥t♦♥ t♥ ♣r♥t rt♦♥ ♥ sttt♦r② t① rts ♦♥ ♦r♥r② ♥♦♠ ♥ ♥ ①♣♥s♦♥ ♦ t r♥ ♥♦♠ t① rt ♦r ss ts ❲ ♥ tt ♦r t♦♥st♥t st② stt ♥②ss t rs♣♦♥s ♦ ♠r♦♦♥♦♠ rts r s♠rr♦ss t① s②st♠s ♦r ♦t ♣♦② ♥s ♦r rt♦♥ ①sts ♥ t ♣ttr♥ ♦♦s♦ t①♣rrr ♦♥s♠♣t♦♥ ♦s ♥ ♦r s♣♣② ♦r s♥ t ①♣t♠♦♥ ♦ t t① ♦ ♣trs ♥trt♦♥ ♠♦♥ ♥r②♥ t① ♣r♦s♦♥s ♥ ttrtrts s♣ t①♣②rs s ♦s♦ ♦r s ss♦t t s♥♥t q♥ttt ♥ qtt r♥s ♥ ♠r♦♦♥♦♠ rts ♦♥ t♥♥tr♥st♦♥ ♣t ♠♠t② ♦♦♥ ♣♦② ♥ s rsts ♥t tt ♥♦♥t♦♥ s♠♦♦t t① ♥t♦♥s ♠② ♥ ♣♣r♦♣rt ♠♦♥ s♠♣t♦♥♦r ♣r♦r♠♥ st② stt ♥②ss t tr♦♥♦s♥t ♠♦s t② r ss st ♦r tr♥st♦♥ ♥②ss ♦ t① ♣♦② ♥s r ♥♥s ♣r♦ s♣♣♦rt ♦r t♥♦r♣♦rt♦♥ ♦ ①♣t t① t ♥ ♦ ♣rs♠♦♥② ♦r ♣r♣♦ss t① ♣♦② ♥②ss♥ ts ss ♦ ♠♦s

♦

♥ ts st♦♥ s♣② rs ♠♦ r ♠rt ♥trt♦♥s t♣ t♥ ♦s♦s r♠s ♥♥ ♥tr♠r② ♥ ♦r♥♠♥t ♥ ♠ t① t♦r t♦ ①♣t② ♠♦ ♣r♦s♦♥s tt tr♠♥ t t①trt♠♥t ♦ ♦r ♥♦♠ ♦s♦s ♠ ♦♥s♠♣t♦♥ s♥ ♦r s♣♣② ♥rs♥t ♦s r♠s r ♦r ♥ r♥t ♣t t♦ ♣r♦ ♦♠♣♦st ♦t♣t ♦♦tt ♥ tr♥s♦r♠ ♥t♦ tr ♦♥s♠♣t♦♥ ♦r rs♥t ♦♦ ♦r ♥♥sst ♥♥ ♥tr♠r② ts ♥ ♣♦sts ♦ ♥♥ ssts r♦♠ ♦s♦s♥ ♦ts t ♥s t♦ ♦♥s♠r ♥ ♠♦rt ♦♥s r ♦r♥♠♥t ♦♥sr♥t ♦s♥ ♣t ♥ ♣r♦t ♣rt s♥ss ♣t r♠tt♥ t rtr♥ ♦♥ts ♣♦rt♦♦ t♦ ♣♦st♦♥ ♦s♦s r stt ♥ ♦ ♦r♥♠♥ts♦t t① ts ♦ ② ♦s♦s ♥ r♠s ♥ ♠ ♦♥s♠♣t♦♥ ①♣♥trs♣ ♣t ①♣♥trs ♥ tt tr♥sr ♣②♠♥ts ❲ ♥ts ♣rt♦rst rr♥ t ♣t ♦♥♦♠ rts ♣rs ♥ s rs ss♦tt ♥ ♣♦② ♦s♦s ♠♦rtt② rs

♦s♦ st♦r s ♦♣ t♦ ①t tr♦♥t② ♦♥ s♣ ♠♥s♦♥ss s tr r ♦ st s ♣rtr r♦♠ ♣rs♠♦♥♦s r♠♦r s♥ssr② s♦ tt t t① t♦r ♥ ①♣t② ♠♦ t sr ♣r♦s♦♥s ❲

♥ ① ♥t tr♦♥t② ♥ ♦r ♣r♦tt② ♠② ♦♠♣♦st♦♥ ♥ t♥♦♠♥ts ♠♣② ① ♣♦st tr♦♥t② ♥ rs♥t ♦ ♥ t♥r s s t ♣t ♦ t♠ ♥♥ t ♥♦r♣♦rt♦♥ ♦ ts t s r ss s♦♥ ♥ t♦♥ ♥r②♥ t① ♣r♦s♦♥s ♥ ♥trt♦♥s tt ♥♥♦s♦ ♦r ♦♦♥ ♣♦② ♥ ♦r t s♣t♦♥ ♦ sr ♣r♠trs♥ trts ♥♥ t♦s ♦r t ♥tr♥ t① t♦r rt♦♥ ♦ t ♠♦ rs② ♦♥ t ♦♥t ♦♠♠tt ♦♥ ①t♦♥s ♥ ① ♦ ♠ss ♦ t r♦♠ ♥ t① rtr♥s t t ♥tr♥ ♥ r ♥♦♠♣ ② t ttsts ♦ ♥♦♠ s♦♥ rt♦♥ ♠t♦♦♦② ssr ♥ t♦♥

①♦♥♦s ♣♦♣t♦♥ r♦t ♥ t♥ ♣r♦rss r ♦t ♣rs♥t ♥ t ♠♦ r♠♦r s tr♦r strtr t♦ ♦♥sst♥t t t ①st♥ ♦ ♥r♦t Pt P ♦♥ ♦r t ♣♣t♦♥ ♦ stt♦♥r② s♦t♦♥ ♠t♦s t♦ ♦♠♣t qr♠ tr♥stt♦♥r② ♦r♠ ♦ t ♠♦ ♦♥sst♥t t P ss♣ s ②♥♠ ♣r♦r♠ ♥ ♣♣♥① rtr♠♦r s♥ ♦s♦s st♦ s♦♥♠♥ rt♦♥s ♥rt s♦♥ rs tt ♥♦♥r♥t♦r s♦♠ ssts ♦ t stt s♣ r② ♦♥ srt stts♣ s♦t♦♥ ♠t♦ ♥t♦♥ trt♦♥ rt s♦t♦♥ ②r ♦rt♠ s t♦ s♦ t ♠♦ ssr ♥ ♣♣♥①

♠♦r♣s

♦♥♦♠② s ♣♦♣t ② J =| J | ♦r♣♣♥ ♥rt♦♥s ♦ ♥t② ♦s♦sr J s t st ♦ ♣♦ss srt ♦s♦ s t ♠①♠♠ ♦ J ♦r♦r♥ s j = 1, . . . , R ∈ J ♥s ♥ ♦s♦s ♦ ♠ t♦ ♦rs ♥ ♦♥s♠ ♦♥ ♦s♥ ♥ ♥♦♥♦s♥ ♦♦s ♥ srs ♥s ♠strtr ② R t ♦♥t♥ t♦ ♠ s♥ ♦s♥ ♥ ♥♦♥♦s♥ ♦♥s♠♣t♦♥♦s ♦r s j = R + 1, . . . , J ∈ J ♦s♦s r ss♠ t♦ sr ♣♦ss♦r♥ s t ♣r♦t② πj = 1 ♥ ♥ t♦ ♠♦rtt② rs ♣♦♥ r♥ t♠①♠♠ rtr♠♥t t t ss♦t ♦♥t♦♥ ♣r♦t② ♦ sr♥ r♦♠ j t♦ j + 1 ♦ 1 > πj > 0 ♥t t ♠①♠♠ ♦ J r πJ = 0 s tt t♦s♦ s t rt♥t②

♥rt♦♥ ♦♦rt ♦ j ♦♥ssts ♦ srt ♦s♦ r♦♣s r♥ ②r②♥ ♦r ♣r♦tt② ♣r♦s ③zj ♥① ② z ∈ 1, . . . , nz ≡ Z ♥ ♠②♦♠♣♦st♦♥ f = s ♦r s♥ ♥ ♥ f = m ♦r ♠rr ♦♣ ♥ ♣r♦ J × nz × 2 st♥t ♦s♦s ♠ ♦♥♦♠ s♦♥s

♥t ♣♦♣t♦♥ s ♥♦r♠③ t♦ ♥t② P0 = 1 ♣♦♣t♦♥ r♦s ①♦♥♦s② t rt υp tt♥ ΥP = (1 + υp) t ♠sr ♦ t♦t ♣♦♣t♦♥ t ♥② t♠ ts Pt = ΥPPt−1 = Υt

P ♠sr ♦ ♦s♦ ♠② ♦♠♣♦st♦♥ f j ♥ ♦r♣r♦tt② t②♣ z s Ωe,f

t,j s♦ tt t ♥② t♠ t t ♣♦♣t♦♥ ♠② r♦♥ ♦♥② t ♠sr ♦ s♥s ♥ ♠rr ♦♣s ② ♥ ♣r♦tt②

Pt =

∫

Z

∫

J

(

Ωz,st,j + Ωz,m

t,j

)

dj dz

❲ t ♣r♦tt②♠② ♠♦r♣ strt♦♥ ♦ ♦s♦s r♠♥s♦♥st♥t ♦r t♠ t ♠sr ♦ ♦♠♥t♦♥ ♦ ttrts r♦s tr♠♥st②t t r♦ss rt ΥP

♦r sr♣t♦♥ ♦ t ♦♥t ♦♠♠tt ♦♥ ①t♦♥s ♥ ① ♦ s

♦s♦s

♦s♦s ♠ ♦♥s♠♣t♦♥ ♦r ♥ rs♥t s♦♥s t♦ ♠①♠③ t ♣rs♥ts♦♥t ♦ tr t♠ tt② s r r♦♠ ♦♥s♠♥ ♥♦♥♦s♥♥ ♦s♥ ♦♦s ♥ srs ♥ ♠♥s r♦♠ ♠rt ♦r ♥t♦♥ ♦r ♦s♦ ♦ j t ♦r ♣r♦tt② t②♣ z ♥ ♠② ♦♠♣♦st♦♥ f t t♠ ts V z,f

t,j (aj, hoj) r ♥♥♥♦♣r♦ ♥♥ ssts aj ♥ ♦s♥ st♦ hoj r

stt rs ♦s♦ ♦♣t♠② ♦♦ss tr ♥♥ ssts aj+1 ♦♥r♦♣ ♦s♥ srs hoj+1 ♥ rr♥t ♠rt ♦r s♣♣② nj t♥ ♥t♦ ♦♥tt ♥trt♠♣♦r ♦s ♦ ♦r♥r② ♦♥s♠♣t♦♥ cij rt ♥ cgj ♥ r♥t

♦s♥ srs hrj ♠①♠③ tr ♥st♥t♥♦s tt② ♥t♦♥ U z,ft,j r♥r②

♦♥s♠♣t♦♥ ♥ ♦t♥ r♦♠ ♦♥s♠♣t♦♥ ♦ ♠rt♣r♦ ♦♦s cMj ♦r ♦♠♣r♦t♦♥ chj

r♥r② ♦♥s♠♣t♦♥ ♥ rt ♥ r ♦♠♣♠♥tr② ♥ ♣r♦ ♥♦♥♦s♥ ♦♥s♠♣t♦♥ ♦♠♣♦st ♦♦ cj ♥r ♦♣ ♦s♥ ♥ r♥t ♦s♥srs r ♣rt ssttts ♥ s♠♠r③ t ♦s♥ sr ♦♠♣♦st ♦♦hsj s ♥ rs ♥ ♦ ♥ r♥s s ♥ t ♦r♠r ♦r ♥tr♣rt♦s♥ srs s t ♦s♦s st♦ ♦ r ♦♦s ♣s t st♦ ♦ rs♥t♣t s ♥ t ttr ♦r ♦s♦s tr♥st♦♥ ♦st ξHj ss♦t t♥♥ rs♥t stts r♦♠ r♥tr t♦ ♦♠♦♥r ♦r rs t♦ ♠t trq♥② ♦ stts ♥s t♦ ♦♠♣♦st ♦♦s r t♠ss ♥st ♥t♦ tr♦♠♣♦st ♦♦ xj ♥ s♦♥ sr lj s ♦t♥ r♦♠ t ♣♦rt♦♥ ♦ ♥tr②t♠ ♥♦♠♥ts ♥♦t s♣♥t ♦♥ ♠rt ♦r ♦r ♥♦♥♠rt ♦r nh

j ♦ ♥♥t③ ♦s♦s t♦ ♠ rt ts ss♠ ❵r♠♦ ♠♦t

♥r♦♥ rt ts r ♠ ♥ tr♠s ♦ ♥ ♦♦s ♥ r ss♠ t♦ r ② ♥ts ♦ts ♦ t ♠♦ s♣t♦♥ ♦ t ♥♦♥♦s♥♦♥s♠♣t♦♥ ♦♠♣♦st ♥s ♦t rt ts ♥ ♦r♥r② ♦♥s♠♣t♦♥s ♦s♥ s t ♦s ♦r s t♦ ♣tr t ♠♣r② ♦sr r ♦rt ♥ ♦r (f, z) ♠♦r♣ t♦ t ♥rstts t r♥

rt ♦r s♣♣② ♦♣rts ♦♥ ♦t ♥ ①t♥s ♥ ♥t♥s ♠r♥ s♦ tt♥s t♦ rt ♠♣♦②♠♥t ♥ r♦♥ ♦♥ ♥t♦ ♠♦♠♥t ♦ ♦rrs ♥t♦ ♦r♦t ♦ t ♦r ♦r s s ♥s ♥ ♦rs ♣r ♦rr ♦r♠r ♦♠♣♦♥♥t s♠♣♦rt♥t s t s ♥ tr♠♥ s t ♣r♠r② rr ♦ rt ♠♣♦②♠♥ttt♦♥s ②♥ ♦rt♦ ♥ ❩♥ ♦ ♦ ♣r♦♠s ss♦tt t rs ♦ ♠♥s♦♥t② ♦♦ ♥ t ♥ s♣② ♥s♠rt ♦r s♣♣② nj ∈ N ≡ 0, nPT , nFT s tt ♥s ♠② ♦♦s t♥♥♦ ♦r ♣rtt♠ ♦r ♦r t♠ ♦r ❲ s♦♠t rstrt ts s♣t♦♥♦s s t♦ ♣tr t ♦srt♦♥ tt ♦r♥ ♦rs t♥ t♦ ♥ r♦♥ ♣rtt♠♥ t♠ s ♥ ♥ ❲s

r♥ r♦♠ t ♦srt♦♥ tt ♠♣♦② ♥s s♣♥ ss t♠ ♦♥ ♦s♦rt♥ t ♥♠♣♦② rr ♥ r ss♠ tt t t♠ ♥ s♣♥s ♦♥ ♦♠ ♣r♦t♦♥ ①♦♥♦s② rs ♥rs② t tr ♦s♥ q♥tt② ♦

♥ ♣♣♥① r♥ t stt s♣ t♦ ♣r♠t ♥♠r ♦♣t♠③t♦♥ ♦r s♥ sttr ♥t ♦rt yj ≡ ho

j + aj ♦r ♦s♦ ♦ ♠♦r♣ (z, f, j) ♥ s♣② hoj ♥ aj s

♦ rs ♥ tt t ♣r♦♠ s s♦ rrs② ② rs ♥t♦♥ s sr ♥♣♣♥① ts ♥ ♦ rs ♦s ♥♦t tr t strtr ♦ t ♣r♦♠ ♣rs♥t r

tr♥t② t ♠② ss♠ tt rt ts r ♠ ♥ tr♠s ♥♥ ssts rt♥ s♣♥t ♦♥ ♥ ♦♦s ② t r♣♥ts s ♦ q♥t t♦ ♦r s♣t♦♥ t s ss♠tt t r♣♥ts r ♥♦♥t① ♥ ♦♣rt ♦stss② st t♦ ♥ ♥tr♣r♦ ♥ t

♠rt ♦r s♦ tt t ♠♦♥t ♦ ♦♠ ♦r s tr♠♥ ② t ♠♦♥t ♦ ♠rt♦r tr♦ t ♥t♦♥ nh

j (nj) ♥ ♦s♦r ♥ r② ♦ts♦r rtrss♠ tt ♦s♦s r t s♠ ♦ ♦♠♣r♦ ♦♥s♠♣t♦♥ r♦♠ ♥♦♥♠rt ♦r ♦r tr♦ t ♥t♦♥ chj (n

hj ) s♣t ts s♠♣ strtr

♦ ♦♠ ♣r♦t♦♥ ts ♥♦r♣♦rt♦♥ ♣s ♥rt ② tr♦♥t② ♥ ♠rt♦r ♦rs r♦ss ♠♦r♣s s ♦sr ② ♥ ♥ ♦③♥♦ rst rr♥♥ ♥s ♥ t ♠♦ t♥ t♦ s♣♣② rt② ♠♦r ♦rs ♥ t ♠rt t♥♦r r♥♥ ♥s ♦♥ s ♦s♦s ♥ ♦♠ ♠♦r ♣r♦t ♥t ♠rt r♥ ♦ t ♥t rtr♥ t♦ ♦r ♥rss ♥ rs rtr rt♦♥ ♥♠rt ♦rs t ♦r s

♦t ♦sts t♦ ♠rt ♦r ♥ tt② ♦st ♠♦♥tr② ♦st ♥ ♦♥s♠♣t♦♥♦st ♦♦♥ ♦tr t s♥ ♦s♦s ① tt② ♦ss ♦ F s t ♥ ♥trs t ♦r ♦r ♠rr ♦s♦s ① tt② ♦ss♦ Fm t s♦♥r② r♥r ♦rs ♠♦♥tr② ♦st κz,fj ♣trs r ♦sts♥r t ♥ s ♥t♦♥ t ♥♠r ♦ q②♥ ♣♥♥ts t♥ tt♦s♦ νz,fj ♥ t ♠rt ♦r ♦rs ♦ t s♥ ♦r s♦♥r② ♦rr s♦st ♦s ♦r t ♠♦ t♦ ♣tr rt♦♥ ♥ ② ♠rt ♦r ♥ ♦r♦♥r♥♥s t♦ rr♥ t ♥② ♥♥ ♦r♥r② ♦♥s♠♣t♦♥s t s♠ ♦ ♠rt ♥ ♦♠♣r♦ ♦♥s♠♣t♦♥ ♦♦s ♦r s♣t♦♥ ♦ t♠s ♦r ♦♠ ♣r♦t♦♥ ♠♣s tt ♦s♦s ♦ss ♥ ♦r♥r② ♦♥s♠♣t♦♥ rs ♣♦st② t ♠rt ♦r ♦rs ♦rs♦♥ ♥ ❲♥s s♦tt s ♦♥s♠♣t♦♥ ♦st ♣s t♦ r♣t t ♠♦st② r♣t ♥tr ♦ rtr♠♥t♦sr ♠♣r② ♣rs♥ ♦ ts ♦st ♥ ♦r ♠♦ ♥ ♥ ♥s ♦r♥t ♠♦r♣s t♦ ♦♦s rtr♠♥t t r♥t s ♣r♦r t♦ t rqrrtr♠♥t rtr ts trs ♣ ♥rt t ♠♣r② ♦sr ①t♥t♥ ♥t♥st② ♦ ♦r ♦r ♣rt♣t♦♥ r♦ss r♥t ♠♦r♣ r♦♣s t♦t♠♣♦s♥ ①♦♥♦s ② rt♦♥ ♥ ♦r stt② s ♦♥

♦♥sr s♥ ♥ ♦t ♦ ts ♦s♦s ♦♣t♠③t♦♥ ♣r♦♠♦r ♥♦♥ ♣♦② r♠ s

V z,st,j (aj, h

oj) = max

aj+1,hoj+1

;

xj ,nj∈N

U z,st,j (xj, nj) + βπjV

z,st+1,j+1(aj+1, h

oj+1)

U z,st,j (xj, nj) ≡ max

hrj ,c

ij ,c

gj

log(xj)− ψsn1+ζs

j

1 + ζs− F s

r

yj ≡ hoj + aj

xj ≡(

σcηj + (1− σ)hsηj)1/η

cj ≡ (cij)θz,s(cgj )

(1−θz,s)

hsj ≡ maxhoj , hrj

❲ ♣rs t①s ♥ tt② r t♠ ♣♥♥t t ♦s♦ ♣s tr ♦ ♦ rs ♦rt♠ s♥ ♦ r ♥♦tt♦♥ ttr ♦♠t t t♠ ssr♣t r ♥t rt♦♥

nj =

1− lj − nhj (nj) ∀j ≤ R

0 ∀j > R

F s =

φs nj > 0

0 nj = 0

cij ≡ cMj + chj (nhj )

♥t♦♥ ♦r♠ ♦r ♥st♥t♥♦s tt② ♥ qt♦♥ s ♦s♥ s t s♦♥sst♥t t P ♥ t ♣rs♥ ♦ ① tt② ♦sts r♦♠ ♦r♥

♦s♦s tr s ♦ st rstrt ② t ♦♥str♥t s♠ ♦ ①♣♥trs ♦♥ ♠rt ♦♥s♠♣t♦♥ cMj rt ♥ cgj ♥ r♥t ♦s♥prth

rj s s t ♦ ♦ t ♥①t♣r♦ st♦ ♦ ♥♥ ssts aj+1 ♥ ♦♥r

♦♣ ♦s♥ hoj+1 ♥ ♥♦ rr t♥ t rs♦rs rr♥t② t♦ t♦s♦ rs♦rs ♦♥sst ♦ t r♦ss rtr♥ ♦♥ ♥♥♥♦♣r♦ ♥♥ssts (1 + rpt )aj ② ♥♥ ♥tr♠r② t trt① qsts r r♦♠t♦s ♦ t t ♥ ♦ t ♣r♦s ♣r♦ beqt t st♦ ♦ ♥♥♥♦♣r♦♦♥r♦♣ ♦s♥ ss ♠♥t♥♥ ♦sts ♥ ♦♥♦♠ ♣rt♦♥ (1− δo)hoj t♦ ♦ ♥♦♠ iz,st,j s q t♦ ♦r ♥♦♠ njwt③

zj r♥ ♦r♥ ②rs ♥ q

t♦ s♦ srt② ♣②♠♥ts ssz,sj r♥ rtr♠♥t s rs♦rs ♠② r ②♥t t① ts T z,s

t,j r ♦sts κz,sj ♥ ♦s♥ tr♥st♦♥ ♦sts ξHj ♦r♠②t t ♦♥str♥t ts t ♦r♠

cMj +cgj +prth

rj +aj+1+h

oj+1 ≤ (1+rpt )aj+beqt+(1−δo)hoj + i

z,st,j −T z,s

t,j −κz,sj −ξHj

r

iz,st,j ≡ njwt③z,sj + ssz,sj

κz,sj = ccz,sνz,sj nj

ξHj =

φohoj+1 hoj = 0

φrhrj+1 hoj > 0

♥ ccz,s s ♥ ①♦♥♦s s ♣r♠tr ♦r ♦st ♣r tr♠♥t♦♥ ♦ ♥tt① ts T z,f

t,j r t ♥ t♦♥ ♦s♦s r ♣r♠tt t♦ ♦rr♦ ♥ ♠t t ♥ ①ss ♦ s♥s st

t♦ t ♦♦♥ rstrt♦♥s

yj ≥

②z,s hoj = 0

γhoj hoj > 0

r ②z,s < 0 s t ♦r♦♥ ♦ t ♥t ♦rt s♣♣♦rt ♥ t ♣r♠tr 0 ≤ γ ≤ 1♥ ♥tr♣rt s t ♦♥♣②♠♥t rt♦ ♦r t ♠♥♠♠ qt② ♦♠♦♥r♠② ♦ ♥ tr ♦♠ s ♦♥t♦♥ ♠♣s tt r♥trs r t♦ ♠t

♦tr rr ♥ t♣♥ ♦r t ♣r♦♦ ♦ ♥ r♦t ♣t ♦♥sst♥②

♥sr t ♣ t♦ t ♠♦♥t ♦ ②z,s ♦♠♦♥rs ♠st ♠♥t♥ ♠♥♠♠ qt②♥ ♦ γhoj ♦♠♦♥rs ss t♦ rr ♦♥s s♥ tr ♦♠ s ♦tr♦r ♦rr♦♥ ♣ t♦ (1− γ)hoj

♦t r♥t ♦s♥ ♥ ♦♥r♦♣ ♦s♥ r st t♦ ♠♥♠♠ s③s rr < o ♠♥ r♥ts rt② ♠♦r ♦r ♠r② ♦r♥r② ♦♥s♠♣t♦♥ ♠st t st s r s ssst♥ i ❲ rtr ss♠ tt ♦s♦s ♦ ♥♦t♠ rt ts ♦r♥r② ♦♥s♠♣t♦♥ s t ts ssst♥

hsj ≥ r, cij ≥ i

hoj ≥ o hoj > 0

cj = cij cij = i

s sss ♥ t♦♥ t r ♦r♥♠♥t ♣r♦ ♦♥t♦♥ rtr♥sr t♦ ♥② ♦s♦ tt ♥♥♦t ♦r ♦t i ♥ r

t s ss♠ tt ♦s♦s ♥tr t ♦♥♦♠② t ♥t ♥♥ ssts ♦ a1 ♥③r♦ ♦♥r♦♣ ♦s♥ ♦ ♦s♦ t♦ t ♠①♠♠ J t② rss♠ t♦ t ③r♦ ♥t♦rt s s ♥♦t t♦ s② tt s♥s aJ+1 q ③r♦ tt ♥ ♦ s ♦♥r♦♣ ♦s♥ ♣t ♦ ♣♦st ♦♥ ♥ts t♦ t ♠♦rt t ♥ ts ② t ♠♦ ♦s ♦r rrs ♠♦rts ♦r♠② ♠♣♦s t ♥t ♥ tr♠♥ ♦♥t♦♥s

y1 = a1

ho1 = yJ+1 = 0

V z,st,J+1 = 0

rr ♦s♦s s♠r ♦♥str♥ ♦♣t♠③t♦♥ ♣r♦♠ t♦ tt ♦ s♥ ♦s♦s ② ♠ ♦s ♦♥ t s♠ ♠r♥s t♦ ♠①♠③ t ♣rs♥ts♦♥t ♦ t♠ tt② t t ♦♦♥ ♦t ♥t♦♥

V z,mt,j (aj, h

oj) = max

aj+1,hoj+1

;

xj ,n1j ,n

2j∈N

U z,mt,j (xj, n

1j , n

2j) + βπjV

z,mt+1,j+1(aj+1, h

oj+1)

t t ss♦t ♥st♥t♥♦s tt② ♥t♦♥ ♦♥ ♦r t♦ ♣♦t♥t r♥rs

U z,mt,j (xj, n

1j , n

2j) ≡ max

aj ,hrj ,h

oj ,

cij ,cgj

log(xj)− ψm,1(n1

j)1+ζm,1

1 + ζm,1− ψm,2

(n2j)

1+ζm,2

1 + ζm,2− Fm

rr ♦s♦s s♠r② qt♦♥s tr♦ ♥① f = m♥st ♦ f = s t t ①♣t♦♥ ♦ qt♦♥s ♥ rstqt♦♥ ♣♣s t♦ ♦r s♣♣② ♦r ♦ t ♠rr ♦s♦s ♣♦t♥t r♥rs♥② s♦ tt n1

j ♥ n2j ♥♦t t ♣r♠r② ♥ s♦♥r② ♦r s♣♣② r

♦rr s ♣ t s♠ q♥tt② ♦ ♦♠ ♣r♦t♦♥ ♦rs t♦ ♦r ♥ t ♠rt

♦♥ s♥ t s ss♠ tt t ♦sts ss♦t t ♦r♥ ♣♣② ♦♥② t♦ ts♦♥r② r♥r

nǫj =

1− lǫj − nhj (n

ǫj) ∀j ≤ R, ǫ = 1, 2

0 ∀j > R

Fm =

φm n2j > 0

0 n2j = 0

κz,mj = ccz,mνz,mj n2j

♦♠ ♣r♦t♦♥ ♥ ♥♦♠ ♣♥ ♦♥ t ♦r ♦ ♦t ♥s ♥ ♦s♦

cij ≡ cMj + ch,2j (nh,1j ) + ch,1j (nh,2

j )

iz,mt,j ≡ (n1j + µzn2

j)wt③z,mj + ssz,mj

r 0 < µz ≤ 1 s ♥ ①♦♥♦s ♣r♦tt② t♥ t ♣r♠r② ♥ s♦♥r②♦rrs ♦t tt t s♦♥r② r♥rs t rt ♣♥s ♦♥ t ♣r♦tt②t②♣ s♣ tr♠ ③z,mj s♦ tr♠♥s t ♣r♠r② ♦rrs t rt♥ t ♣r♥ ♠rt r rt wt s s♣t♦♥ s ♥t♥ t♦ ♣tr♦t t ♦sr r♥♥s r♥t ♦ ♦rrs t♥ ♠rr ♦s♦ ♥ ♣♦stss♦rtt ♠t♥ r♥♦♦ t t

♥② s♥ ♦s♦s ♥ ♥①♣t② ♥ R < j < J t② ♠② ♥ t tt t② ♠t ♦r tr t♠ ♦ ♦s♦ ♦r r♥ t ♠①♠♠ J Λ ♣r♦♣♦rt♦♥ ♦ tr ♥t ♦rt s ♦t t♦♥♦ ♦♥s♠♣t♦♥ ①♣♥trs ♥ t r♠♥♥ (1 − Λ) ♣r♦♣♦rt♦♥ s ♦stss②qt ♥ ♦t ② t ♦r♥♠♥t t① ♥ rstrt ♥ ♠♣s♠s♦♥ ♠♦♥ t ♥ ❲ ts s♣t♦♥ s ♦s♥ ♣r♠r② s♦ tt t ♦sr ♦ rt qsts ♥ trt tr♦ t ♦ Λ t s♦ ♣trst ♦sr ♥rs ♥ ♠ ①♣♥trs t t ♥ ♦ r♥ t rtr♠♦r ♦r ♣rt♦rst ss♠♣t♦♥ ♠♣s tt ♥ts ♥ ♣rt tq♥tt② ♦ t t ♥ ② t ♦♥ ♦r s t♦ s♣② ♦♥t♠♣♦r♥♦srstrt♦♥ ♥ t ♦♦♥ ♠♥♥r

beqt = (1− Λ)

∫

Z

∫

J

(1− πj)∑

f=s,m

yt+1,j+1Ωz,ft,j dj dz − T beq

t

r T beqt s t rt ♠♦♥t ♦ t①s ♦t ② t ♦r♥♠♥t ♦♥ ♥t

qsts t t t ♥ ♦ ♣r♦ t

r♠s

♥t r♠s r ♦r rt② r♦♠ ♦s♦s ♥ ♣rt② ♦♠♣tt ♦r ♠rt♥ r♥t ♣t r♦♠ ♥♥ ♥tr♠r② t♦ ♣r♦ ♥ s ♦t♣t ♥ ♦♠♣tt♦♦s ♠rt t ♣r♦t ♠①♠③♥ s Pr♦t♦♥ t♥♦♦② s ss♠ t♦ ♦ t♦♦s ♦r♠

Yt = GgtK

αt (AtNt)

1−α−g

r Gt = Gfedt +Gsl

t t s♠ ♦ ♥♥♥♦♣r♦ ♣ ♣t ♦♥ ② t r♦r♥♠♥t s s t stt ♥ ♦ ♦r♥♠♥ts Kt ♥ Nt r ♥♥♥♦♣r♦♣r♦t ♣rt s♥ss ♣t ♥ t ♦r ♥ts s ♥ ♣r♦t♦♥ ♥ At

s t ♦ ♦r♠♥t♥ t♥♦♦ ♣r♦rss ♦s s At+1 = ΥAAt r ΥA = (1+υA) s t ①♦♥♦s ♥♥ r♦ss rt ♦ t♥♦♦ r♦t ♥♦t♣t ♦♦ Yt s t ♥♠érr ♥ ♥ ♦stss② tr♥s♦r♠ ② ♦s♦s ♥t♦ ♦♥s♠♣t♦♥ ♦♦ ♦♥r♦♣ ♦s♥ srs ♦r ♥♥ sst s ♥ rs r♥á♥③❱r ♥ rr ♥ ♦ ♥ r♥s ♦r ♥t♦ rt t

rt t ♦r Nt s t s♠ ♦ ♥②t ♦r ♦rs

Nt =

∫

Z

∫

J

③z,sj nz,st,jΩ

z,st,j + ③z,mj (nz,1

t,j + µznz,2t,j )Ω

z,mt,j dj dz

r Ωz,st,j s t ♠sr ♦ ♣r♦tt② ♥ r♦♣ ♦r s♥s t t♠ t Ωz,m

t,j st ♦rrs♣♦♥♥ ♠sr ♦r ♠rr ♦s♦s ♥ ♦r ♦rs ♦r s♥s ♠rr♣r♠r② ♥ ♠rr s♦♥r② r♥rs r nz,s

t,j nz,1t,j ♥ nz,2

t,j rs♣t②❯♥r ts ♥r♦♥♠♥t t ♣r♦t♦♥ st♦r ♥ ♠♦ r♦♠ t ♣rs♣t ♦

r♣rs♥tt r♠ s t ♦t t♦ ♦♦s t q♥tt② ♦ ♥♣ts Kt ♥ Nt

t t♦r ♣rs rt ♥ wt t♦ ♣r♦ ♦t♣t Yt t tt ♠①♠③s trt① ♣r♦ts ♣r♦ t♥ t st♦ ♦ ♣ ♣t Gt s ♥ ss♠♥ tt ①♣♥ssr ② t r♦♠ s♥ss t①s ♣r♦ts Πt ♦r t r♣rs♥tt r♠ r♥ ②

Πt = maxKt,Nt

(1− τ bust )(1− τ slbt )(

GgtK

αt (AtNt)

1−α−g − wtNt

)

+ τ bust lsdbust − rtKt

r τ bust ♥ τ slbt r ♥r s♥ss t① rts ♣♣ t♦ t① ♣r♦ts t tr ♥ t ♦♠♥ stt♦ rs♣t② ♥ lsdbust s ♠♣s♠t♦♥ ♥st t r t① ts tt ♥ ♦ s♥ss t①s rss♠ t♦ ② t r♦♠ r s♥ss t① ts

❲ t ♣rs♥ ♦ ♣ ♣t s rs t♦ ♦♥♦♠ r♥ts ♣rt ♦♠♣tt♦♥♥ t ♦♦s ♠rt ♠♣s tt r♠s r♥ ③r♦ ♦♥♦♠ ♣r♦ts ♥ qr♠ ♦♦♥t ♦r ts ss♠ tt t ♥♥ ♥tr♠r② s s♥t ♠rt ♣♦r♦r t r♠ t♦ ①trt ts r♥t rs t♦ t ♦♥rs ♦ ♣t r♠s t♥ r♣rt t♦rs t t ♥ ♣rs s tt

wt = (1− α− g)GgtK

αt (AtNt)

−α−g

rt =1

Kt

(

(1− τ bust )(1− τ slbt )(α + g)(GgtK

αt (AtNt)

1−α−g) + τ bust lsdbust

)

♥♥ ♥tr♠r②

r♣rs♥tt ♥♥ ♥tr♠r② ♣♦♦s t ♥t st♦ ♦ s♥s ♦s♥ ② ♦s♦s ♥t♦ ♣♦sts t♦ ♥st ♥ ♣t ♠rts ♥♥♥♦♣r♦ rt♣♦sts r ①♣rss s

Dt =

∫

Z

∫

J

∑

f=s,m

az,ft,j Ωz,ft,j dj dz

♣r♦ t ♥tr♠r② ♣②s ♦s♦s t ♣r♥♣ ♣s ♣♦rt♦♦ rtr♥ ♦♥tr s♥s ♦s♥ ♥ t ♣r♦s ♣r♦ (1 + rpt )Dt ♥ ts ♥ ♥ ♣♦sts Dt+1♦r t s ♥ ♥st♠♥t ♦t♦♥ ♥♥ ♥tr♠r② ♥ ♥st ♥♣r♦t ♣rt s♥ss ♣t Kt+1 r♥t ♦s♥H

rt ♥ ♦r♥♠♥t ♦♥s Bt+1

qt♦♥s ♦ ♠♦t♦♥ ♦r ts st♦s r s ♦♦s

Kt+1 = (1− δK)Kt + IKt − Ξt

Hrt = (1− δr)Hr

t−1 + Irt

Bt+1 = Bt +NBt

r s ♦♥① ♣t st♠♥t ♦st Ξt ♦r t♥ r♦♠ t ❵r♥ ♦ ♥st♠♥t tt ♦ ♣ r♦tst ♣rt ♣t ♣r ♣t ♦♥st♥t

Ξt =ξK

2(Kt+1

Kt

−ΥPΥA)2Kt

t ♥st♠♥t ♥ s♥ss ♣t ♥ r♥t ♦s♥ ♦s♥ ② t ♥♥ ♥tr♠r② r IKt ♥ Irt ♥ ♦♥ sss ② t ♦r♥♠♥t ♥ ♣rs ② t♥tr♠r② r ♥♦t NBt ❲ ♥st♠♥t ♥ s♥ss ♣t ♥ ♦r♥♠♥t♦♥s t t♠ t ②s rtr♥s (rt+1 + δK) ♥ ρt+1 t t♠ t + 1 ♥st♠♥t ♥ r♥t♦s♥ s ss♠ t♦ ♠♠t② t♦ ♦s♦s ♥ ts ②s ♦♥t♠♣♦r♥♦s rtr♥ prt ♥tr♠r② ♥rs ①♣♥ss r♦♠ t ♦♥♦♠ ♣rt♦♥ ♦tr st♦s ♦ s♥ss ♣t ♥ r♥t ♦s♥ ♣t s s ♣♦t♥t s♥ss♣t st♠♥t ♦st ①♣♥ss s♠♠r③s t ssts ♥ ts ②t ♥tr♠r② t t ♥ ♦ ♣r♦ t s s tr ♦♥t♠♣♦r♥♦s ♥♦♠ ♥①♣♥s ♦s ♦r t ♣r♦

♦t ♦ t ♥♥ ♥tr♠r② s t♦ ♦♦s t sq♥ ♦ s♥ss ♣t♥ r♥t ♦s♥ ♠①♠③ tr ♥t s ♦

It = maxKs+1,Hr

s

∞∑

s=t

(1 + rs − δK)Ks + (1− δr)Hrs−1 + prsH

rs + (1 + ρs)Bs +Ds+1

− (1 + rps)Ds −Ks+1 − Ξs −Hrs − Bs+1

st t♦ t rs♦r ♦♥str♥t

Ds+1 ≥ Ks+1 +Hrs +Bs+1 ∀s

♥ t qt♦♥s ♦ ♠♦t♦♥ ♦r ♣t st♦s qt♦♥s tr♦ t KsBs ♥ H

rs−1 ♥

rs♦r ♦♥str♥t stts tt s♥ss ♣t ♥ ♦r♥♠♥t ♦♥s tt ♥ ♦ ♥② ♥ ♣r♦ ♣s rr♥t r♥t ♦s♥ st♦ ② t ♥tr♠r②♠st ss t♥ ♦r q t♦ t ♥♦♣r♦ st♦ ♦ s♥s ♣♦st ② ♦s♦s♥ t ♦t ♦ t ♥♥ ♥tr♠r② t ♣r ♦ r♥t ♦s♥ s st s♦ ttt ♥tr♠r② s ♥r♥t t♥ ♥st♥ ♥ r♥t ♦s♥ ♦r s♥ss ♣t ②s t ♥♦rtr ♦♥t♦♥

prt = rt+1 − δK + δr −

(

∂Ξt

∂Kt+1

+∂Ξt+1

∂Kt+1

)

❲ ss♠ tt ♦r♥♠♥t ♦♥s r rt② ② t ♥tr♠r②♥♦② ♦ s ♥trst rt ρt rt t♦ t rtr♥ ♦♥ ♣r♦t ♣t ♥tr s ♥♦ ♥rt♥t② ♥ ts ♠♦ ♦♠♣s ts ② ♥tr♦♥ ♥ ①♦♥♦st♠♥r♥t t♥ ρt ♥ rt

ρt ≡ rt −

♥② ♥♦t♥ t ♦♥t♠♣♦r♥♦s ♥♦♠ ♥ ①♣♥ss t t♠ t r♦♠ t ♥♦♠stt♠♥t ♦ t r♣rs♥tt ♥♥ ♥tr♠r② ♦ t ♠♣♦st♦♥ ♦ ③r♦♣r♦t ♦♥t♦♥ ♦♥ t ♥♥ ♥tr♠r② ♠♣s tt ♦s♦s r rtr♥ ♦♥tr ♣♦sts q t♦

rpt =(rt − δK)Kt − Ξt + prtH

rt − δrHr

t−1 + ρtBt

Dt

♦r♥♠♥t

♦s♦ ♥♦♠ ①t♦♥

♥ ts st♦♥ t t t① trt♠♥t ♦ ♦s♦ ♥♦♠ ♥♦s ts♣t♦♥ ♦ r ♦r ♥♦♠ t① ♣t ♥♦♠ t① ♣②r♦ t① t s♣t① trt♠♥t ♦ s♦ srt② ♥ts s s stt ♥ ♦ t①s ❲ s♣②t ♥r r♠♦r ♦ ♦r ♥♦♠ t①t♦♥ ♥r t ♥tr♥ t① t♦r ♦♣ ♥ ts ♣♣r ♥ ♦r ♣r♣♦ss ♦ ♦♠♣rs♦♥ ♥r t é♥♦ ① ♥t♦♥ é♥♦ ♥ t ♦ ♦ ts ♣♣r t♦ ♠♦♥strt t ♠♣♦rt♥ ♦t t① t ♥♦r♣♦rt ♥ t ♦r t t① trt♠♥t ♦ ♦r ♥♦♠ s ts♠ r♠♦r ♦r ♠♦♥ ♦tr t①s rrss ♦ tr t ♦r t s s ♥ s♠t♦♥

♦ ♦♥rt t ♦s♦s ♦♥♦♠ ♥♦♠ ♥t♦ ♦rrs♣♦♥♥ t① ♥♦♠♦♥♣t ❵rt♦♥ rt♦ s ♥tr♦ t♦ rt t ♣♦rt♦♥ ♦ ♣rtr ♦ ♦♦♥♦♠ ♥♦♠ ♠② st t♦ t①t♦♥ ♣r♦tt② t②♣ ♠② ♦♠♣♦st♦♥ s♣ rt♦♥ rt♦ ♦r ♦r ♥♦♠ χi,z,f ♥ rt♦ ♦r ♣t ♥♦♠ χa♥sr t ♦rrt t① s ♦s♦s st r♦ss ♦r ♥♦♠ iz,ft,j ♥ st

r♦ss ♣t ♥♦♠ rpt az,ft,j r ♦t♥ ② ♣♣②♥ t ♦rrs♣♦♥♥ rt♦♥ rt♦

s tt

iz,ft,j ≡ χi,z,f iz,ft,j

az,ft,j ≡ χaaz,ft,j

qt♦♥ s♠♠r③s t t① t② ♦r ♥ ♦s♦ t t① t②T z,ft,j s q t♦ t①s ♦ ♦♥ ♦r ♥♦♠ taxz,ft,j ♠② tr♠♥ tr

② t ♦r t ♣s t① t② ♦♥ ♣t ♥♦♠ τat rpt a

z,ft,j ♣s t①

ts ss♦t t t ♦ rt② s②st♠ ♦r rtrs τ prt,j iz,ft,j ss r tr♥sr

♣②♠♥ts trsz,ft,j ♣s stt ♥ ♦ t① ts sltz,ft,j

T z,ft,j = taxz,ft,j + τat r

pt a

z,ft,j + τ prt,j i

z,ft,j − trsz,ft,j + sltz,ft,j

❲ ♥♦ sr t s♣t♦♥ ♦ s ♥str♠♥t

①t♦♥ ♦ ♣t ♥♦♠ ♥ tr♠♥t r t① rts ♦♥ ♣t ♥♦♠r tr♠♥ ② r♦♣ ♠② ♦♠♣♦st♦♥ s♣ t① ♥t♦♥s r s ♥q♥t♦♥ st♠t ♦r ♦r♥ s♥ ♦r♥ ♠rr rtr s♥ ♥ rtr♠rr ♦s♦ ❲ ♦♥sr ♦r♥ ♦s♦s s♣rt② r♦♠ rtrs t♦ ♠♦rrt② ♣tr t ♦sr r♥t ♥ ♣t ♥♦♠ t① ts s♥ ♦rrst♥ t♦ s tr♦ t①rr s♥s s ❲ ss♠ tt ♦s♦s rt① rt ♦♥ ♣t ♥♦♠ s ♠♦♥♦t♦♥② ♥rs♥ ♥t♦♥ ♦ tr sst ♦♥srt t♦ t sst strt♦♥ f(at|f, j) s ♦♥t♦♥ ♦♥ ♠② ♦♠♣♦st♦♥♥ tr t ♦s♦ s ♦ ♦r♥ ♦r rtr

τat = q(

az,ft,j ,f(at|f, j))

❲ ♦♦s ts s♣t♦♥ ♦r ♦♥ tt ♣♥s ♦♥ ♦s♦s ♣rtr♠♥ ♣r♦tt② t②♣ s sst ♥♦♠ t♥s t♦ r r♥ ♦r t ② rss t ♣♦t♥t ♦r ♦s♦ t♦ ♠♦ ♥ ♥ ♦t ♦ sr r♥t sst ♥♦♠q♥ts ♦r tr t♠

❲♦r♥ ♦s♦s ♣② ♥t♦ t ♦ rt② ♣r♦r♠ t ♣r♦♣♦rt♦♥ ♣②r♦ t①rt ♦♥ ♦r ♥♦♠ ♣r♦ ♣♣s t♦ t① ♦r ♥♦♠ ♣ t♦ s♣ trs♦ tr ♦s♦s ♣② ♣r♦♣♦rt♦♥ t① ♦♥ tr r♣ts ♦ ♦rt② ♥♦♠ ♣♥s ♦♥ t ♦ ♦ rt② ♥♦♠ ts ♦r♠②

τ prt,j =

p(

iz,ft,j

)

j ≤ R

r(

ssz,ft,j

)

j > R

♥② ♦ ♦r t①s ♦♥ ♥t qsts t ② s ♦s♦s ♥♥t qsts r rstrt ♥ ♠♣s♠ s♦♥ ♠♦♥ ♥ ♦s♦s s♣② tt t t① rt τ beqt s ♥r ♥ qsts ♥ ♥rt t♦ tr t ♥t♦r♦r ♥r② ♦s♦s ♦tr ♥♦♠

①t♦♥ ♦ ♦r ♥♦♠ ♥tr♥ ① t♦r ❯♥r t t① s②st♠♦s♦ t① t② ♦♥ ♦r ♥♦♠ taxz,ft,j s tr♠♥ ② ♣♣t♦♥ ♦ sttt♦r②♠r♥ t① rt s t♦♥s ♥ rts s ♠♣♣♥ r♦♠ ♦ rsstt rs ♥ ♠♦r♣ rtrsts t♦ t① t② s ♦♣ t♦ s♦s t♦ t t s ♣♦ss ♦r t ♣r♦s♦♥s ♠♦ t ♠r♥ t①rt ♦♥ ♦r ♥♦♠ s tr♦r ♥♦t t sttt♦r② t① rt t t ♠r♥ t② ♦♥♥r♠♥t ♦r ♥♦♠ tr ts t♦♥s ♥ rts ♥ ♣♣

r t① rt ♦♥ ♦r ♥♦♠ ♦r t① rts τ it s tr♠♥ ② tsttt♦r② t① rt s ♥ t t① t♦r st r♦ss ♦r ♥♦♠ iz,ft,j st

r♦ss ♦r♥r② ♣t ♥♦♠ krdz,ft,j ♥ t♦♥s dedz,ft,j t♦♥s r ♥t♦♥ ♦st r♦ss ♦r ♥♦♠ st r♦ss ♦r♥r② ♣t ♥♦♠ ♥ t①♣rrr♦♥s♠♣t♦♥ ♦s ♠ ② t ♦s♦ rts crdz,ft,j r ♥♦t ♦♥② ♥t♦♥♦ ♦r ♥♦♠ ♥ ♠② ♦♠♣♦st♦♥ t ♦ ♦tr t① rs s t♦ tr♥t② ♦r tr♦ ♦ r♦s rts ♦r♠②

♥ ♦s♦s r t s♠ rtr♥ ♦♥ tr s♥s rpt ♥ t s♠ rt♦♥ rt♦χa ♥ ♦rr♥ ♦ ♦s♦s ② tr ♥♥ sst ♦♥s s q♥t t♦ ♥ ♦rr♥ ② t t①♥♦♠ r♦♠ ts ssts

taxz,ft,j = max

τ it iz,ft,j , 0

− crdz,ft,j − traz,ft

τ it = τ (iz,ft,j + krdz,ft,j − dedz,ft,j )

krdz,ft,j = k(rpt az,ft,j )

dedz,ft,j = d(iz,ft,j , krdz,ft,j , h

ot,j, c

gt,j)

crdz,ft,j = c(iz,ft,j , krdz,ft,j , κ

z,ft,j , ded

z,ft,j )

traz,ft

6= 0 nj > 0 ♥ f = s

6= 0 n1j > 0 ♥ f = m

= 0 ♦trs

r ♦ ♠♣ss ♥♦ts ♥t♦♥ st tr♠ ♥ qt♦♥ s ♣r♦tt② t②♣ ♠② ♦♠♣♦st♦♥ s♣ tr♥sr ♣②♠♥t traz,ft s s s ♥♦♥st♦rt♦♥r② ♠t♦ ♦ ♥sr♥ tt ♦s♦s t♥ ♥ (z, f) ♠♦r♣r♦♣ ♦♥ r trt r t① rt ♦♥ ♦r ♥♦♠ s tr♥sr ♠② ♣♦st ♦r ♥t ♦r r♥t ♦s♦ r♦♣s ♥ s ♦♥② ♥♦♥③r♦ ♦r ♦r♥ ♦s♦s ❯s♥ qt♦♥ ♥t♦ qt♦♥ s ♦s♦ t① ts ♥ ttrt♠♥t ♦ ♦r t①t♦♥ s s s♣ ♥ t t① s②st♠ ❯♥r ts t① s②st♠♦s♦s ♦♥sr t t① ♠♣t♦♥s ♦ tr r③ ♣t ♥♦♠ ♥ ♠♥tr ♦♥t ♦r s♣♣② ♥ s♥s s♦♥s

①t♦♥ ♦ ♦r ♥♦♠ é♥♦ ① ♥t♦♥ s st♦♥ srs t t① s②st♠ ♦r ♦r ♥♦♠ tt s s ♥♠r ♦r ♦♠♣rs♦♥ t♦ t s ♦♠♠♦♥②s t① ♥t♦♥ ♥r t t♦t t ♦tr t tt ♥rts s♠♦♦t r t① rts ♥ t ♠r♥ t①rts ♦r ♥♦♠ ♦tr ♦♠♠♦♥②s t① ♥t♦♥s t s ♦♥t♥♦s②r♥t ♦s ♦r ♥t r t① rts t♦ ♣tr t t ♦ r♥t① rts ♥ s s② ♣r♠tr③ t t ①♦♥♦s s♣t♦♥ ♦ ♥ t♠r♥ t① rt ♥ r t① rt t t sr ♦ rt♦♥

♦r ♥♦♠ t① ts ♥r ts t① s②st♠ ts t ♦r♠

taxz,ft,j =(

iz,ft,j − λf1 (iz,ft,j )

1−λf2

)

− traz,ft

r t rt tr♠s r t λf1 ♥ λf2 r ♣r♠trs t♦tr tr♠♥ t ♥♦♠t r t① rt ♥ t ♠r♥ t① rt ♣♣ t♦ ♦r♥♦♠ t ♦r ♠② ♦♠♣♦st♦♥ ♥ traz,ft tr♥srs s s ♥♦♥st♦rt♦♥r②♠t♦ ♦ ♥sr♥ tt ♦s♦s t♥ ♥ (z, f) ♠♦r♣ r♦♣ ♦♥ r trt r t① rt ♦♥ ♦r ♥♦♠ st t♦ t ♦♥t♦♥s ♥ qt♦♥ ❯s♥ qt♦♥ ♥t♦ qt♦♥ s ♦s♦ t① ts ♥r t t① s②st♠ ♦r ♦r ♥♦♠

tr ①♠♣s ♦ s♠♦♦t t① ♥t♦♥s ♥ str② ♥ ♦ ♦ ♥ trss ♥ rt ♥ r t

r ♦r♥♠♥t

r ♦r♥♠♥t ♦ts t①s r♦♠ ♦s♦s ♥ r♠s t♦ ♥♥ ♥♦♥♦r♥♠♥t ♦♥s♠♣t♦♥ ①♣♥trs Cfed

t ♥st♠♥t ♥ ♣r♦t ♣ ♣tIfedt s♦ srt② ♣②♠♥ts ♦r rtrs ♥ ♦tr tr♥sr ♣②♠♥ts t♦ ♦s♦st ts r ♥♥ tr♦ t s ♦ ♦♥♣r♦ ♦♥s t t ♥trst rt ρt♦t ♦r♥♠♥t ①♣♥trs ♠st ss t♥ ♦r q t♦ t♦t t① r♥ ♥t ♦tr♥sr ♣②♠♥ts T fed

t ♣s ♥ t sss ss ♥trst ♣ ♦♥ ♦ t

Ifedt + Cfedt ≤ T fed

t +Bt+1 − (1 + ρt)Bt

r Bt s t ♥♥♥♦♣r♦ st♦ ♦ ♦tst♥♥ ♦♥♣r♦ ♦♥s ♦t rt① r♣ts T fed

t r q t♦ t s♠ ♦ rr♥t rt r ♥t t① r♣ts r♦♠♦s♦s T hh

t t①s ♦t ♦♥ ♥t qsts T beqt ♥ t rr♥t r t①

r♣ts r♦♠ r♠s T bust s♦ tt T fed

t ≡ T hht +T beq

t +T bust ♦ ♠♦t♦♥ ♦r r

♣ ♣t s

Gfedt+1 = (1− δg)Gfed

t + Ifedt

t ♦♥str♥t ♥ qt♦♥ ♠♣s (i) ♥ ♠r♦♦♥♦♠ st② sttt ♦r♥♠♥t r♦s ♦r ♦♥st♥t ♦ t s♦ tt ♦♥② ♥♥ rs ρB r♣ ②r ♥ (ii) r♥ tr♥st♦♥ ♣ts ♦r♥♠♥t t ♠② r♦ ♦r sr♥♦ r ♦t ①♣♦s t ♣ts ♠♥t♥ t ♥♦P♦♥③ ♦♥t♦♥

limk→∞

Bt+k∏k−1

s=0(1 + ρt+s)= 0

♠♣s tt t rr♥t st♦ ♦ t s q t♦ t ♣rs♥ts♦♥t ♦ tr ♣r♠r② sr♣ss ♦♥ ♥② qr♠ ♣t

tr ♦s♦s t ♥ ♥♥ s♦ srt② ♣②♠♥t ssz,ft,j r♦♠ t r ♦r♥♠♥t s ss♠ t♦ ♥t♦♥ ♦ t① r r♥♥s ♦r♦r♥ ②rs ♦r tr ♣rtr ♣r♦tt② t②♣ ♠② ♦♠♣♦st♦♥ r♦r♥♠♥t s♦ t③s tr ♦tr tr♥sr ♥str♠♥ts rst ♦s♦s ♥♦r♠②r ♠♣s♠ tr♥srs trlt r ♥t♥ t♦ ♦♥t ♦r s♣ sr ♦rr♥t r ♦r♥♠♥t tr♥srs ♦♥ t ♦r♥♠♥t s ♠♣s♠ t① lstt♦♥ ♦s♦s t♦ ♥♦r♣♦rt t♦s t①s s♣ t♦ ♦♥② ♥ ♥♦♠ t ♥②t♦ ♥sr tt tr ①sts s ♦ st ♦r t rr♥t ♥t ♦rt stt s♣♥ t ♣rs♥ ♦ ♦r ♦♥ ♦♥str♥ts ♦♥ ♦s♥ ♥ ♦r♥r② ♦♥s♠♣t♦♥ s♣② ♦♥t♦♥ tr♥srs trwz,f

t,j ♦r t ♣r♣♦ss ♦ r ♥ ♦s♥ ssst♥ t♦♦♥♦♠ ♦s♦s tt♥ t rt♥ s ♦ t ♦s♦ t ♦♥str♥t♥ qt♦♥ ♥♦t t bdgtz,ft,j

trwz,ft,j = max

0, i + prtr − bdgtz,st,j

s ♠♥s tst rqrs ♦s♦ ♦t t♦ ♥ t♦ ♦r t ♦♥s♠♣t♦♥ ♥♦s♥ ♠♥♠♠s s s ♥♦♥♣♦st ♥t ♦rt ♥ tr ♥①t ②r ♦ ♥♦rr t♦ r♣♥t ♦ r tr♥srs

r♦♠ ♥ t ♣rs♣t ♥t ♥♦♠ t①s ♦t ② t r ♦r♥♠♥t r♦♠ ♦s♦s T hh

t ♦♥sst ♦ ♦r ♥ ♣t ♥♦♠ t① ts ♣②r♦ ♥rtr♠♥t ♥♦♠ t① ts ss tr♥srs ♥ s♦ srt② ♣②♠♥ts

T hht =

∫

Z

∫

J

∑

f=s,m

(

taxz,ft,j + τat rpt a

z,ft,j + τ prt i

z,ft,j − trsz,ft,j − ssz,ft,j

)

Ωz,ft,j dj dz

r

trsz,ft,j = trwz,ft,j + trlt − lstt

r♦♠ qt♦♥ ♥ t tt② ♦ stt ♥ ♦ t①s s♥ssr t① ts T bus

t t t ♦r♠

T bust = τ bust

(

(1− τ slbt )(Yt − wtNt)− lsdbust

)

r ♦r♥♠♥t s ss♠ t♦ ♦t ♥t qsts t t ♥ ♦ ♣r♦ ♥ rstrt t♠ ♥ ♠♣s♠ s♦♥ t ♦♦♥ ♣r♦ tr ♣♣②♥ t① ①s ♦t ♦♥ ♥t qsts t t t ♥ ♦ t ♣r♦s ♣r♦ rtr♦r

T beqt = τ beqt (1− Λ)

∫

Z

∫

J

(1− πj)∑

f=s,m

yt+1,j+1Ωz,ft,j dj dz

tt ♥ ♦ ♦r♥♠♥t

①t♦♥ t ♦r s ♦ ♦r♥♠♥t s ♥♦r♣♦rt ♦t t♦ ♦♥t ♦r tr st♦rt♦♥r② ♣r♦♣rts ♦r tr tt② r♦♠ ♦s♦s ♥ r♠s r t① ts ♦r s♠♣t② ss♠ tt ♦s♦ t① ts ♦ t t stt ♥♦ ♥ ①♣rss s ♣r♦♣♦rt♦♥ ♦ t① ♦r ♥♦♠ ♥ ♦♥r♦♣♦s♥

sltz,ft,j ≡ τ slt iz,ft,j + τ slpt hot,j

r τ slt s ♥r t① rt t♥ t♦ r♣rs♥t ♣♦t♥t② t stt ♥ ♦♥♦♠ ♥ ss t① ♥ τ slpt s ♥r r t① rt ♦♥ ♦♥r♦♣ ♣r♦♣rt②rt ♦s♦ stt ♥ ♦ t① ts ♥ t♥ ①♣rss s

T slht ≡

∫

Z

∫

J

∑

f=s,m

sltz,ft,j Ωz,ft,j dj dz

rt s♥ss stt ♥ ♦ t① r♣ts r

T slbt = τ slbt (Yt − wtNt)

r τ slbt s t stt ♥ ♦ s♥ss t① rt ♥ t s♦♥ tr♠ s s♥ss ♣r♦tsr♦♠ qt♦♥ ♦t stt ♥ ♦ t①s T sl

t r q t♦ t s♠ ♦ t①s ♦tr♦♠ s♥sss ♥ ♦s♦s T sl

t ≡ T slbt +T slh

t s r♣ts r ss♠ t♦ s♣♥t♦♥ ♥♦♥ stt ♥ ♦ ♦♠♣♦st ♦r♥♠♥t ♦♥s♠♣t♦♥ ①♣♥trs Csl

t ♥♥st♠♥t ♥ ♣r♦t ♣ ♣t Islt st t♦ ♥ ♥tr♣r♦ ♥t♦♥t♦♥

Islt + Cslt = T sl

t

r t ♦ ♠♦t♦♥ ♦r stt ♥ ♦ ♣ ♣t s

Gslt+1 = (1− δg)Gsl

t + Islt

qr♠

♥ ♦rr t♦ ①t ♥ r♦t ♣t ♥ st② stt qr♠ t ♠♦ ♠st tr♥s♦r♠ ♥t♦ tr♥ stt♦♥r② ♦r♠ tr♥s♦r♠t♦♥ s sr ♥ ♣♣♥① ♥ t ♠♦ s rs♣ s stt♦♥r② rrs ②♥♠ ♣r♦r♠ ♥ ♣♣♥① qr♠ ♦r ♥ t① s②st♠ s ♦r♠② ♥ ♥ ♣♣♥① s ♦t♦♥ ♦♦s♦ s♦♥ rs tt ♠①♠③ ♦s♦s tt② st t♦ ♦s♦ t♦♥str♥ts ♦t♦♥ ♦ ♦♥♦♠ rts tt r ♦♥sst♥t t ♦s♦ ♦r ♥ t ss♦t ♠sr ♦ ♦s♦s ♣r♦t♠①♠③♥ ♦r ② r♠s st ♦ ♣rs tt tt r♥ ♥ t♦r sst ♥ ♦♦s ♠rts ♥ ♥ ss♦tst ♦ ♣♦② rts tt r ♦♥sst♥t t ♦r♥♠♥t t ♦♥str♥ts ♥ st② stt qr♠ ♠r♦♦♥♦♠ rts r r♦♥ t ♦♥st♥t rt qt♦ t s♠ ♦ t♥♦♦ ♥ ♣♦♣t♦♥ r♦t rts

s♥ rt♦♥

❲ s♣② tt srt t♠ ♥ t ♠♦ ♣sss t ♥ ♥♥ rq♥② st ♦♣r♠trs t♦ rt ♥ ♦t ♥♦♥t① ♥ t① ♣♦② ♣r♠trs ♦t ♦ r② ② ♦♥ s ♦ t ♦♥t ♦♠♠tt ♦♥ ①t♦♥s ♥ ① ♦ ♦r s♣t♦♥ ♠s s ♦ t r♦♠ ♥ t① rtr♥s tt ♥tr♥ ♥ r ♥ ♦♠♣ ② t ttsts ♦ ♥♦♠ s♦♥ ❲ r② t s ♦ ♦♥r♥ st♦r t r♥t ♦srt♦♥s ♥ ♣r♦t♦♥st♦ ♦♥strt ♣r♠tr s ♥ trt♥ t ♣rs♥t ♦♥♦♠ ♥r♦♥♠♥t ♥ t① s ♦s② s ♣♦ss ♦r t ♥t st②stt s♥ qr♠ ss♠t♦ ♦rrs♣♦♥ t ♥r ②r ❲ tt♠♣t t♦ ♠ t ♥t st②stt s♦♠♣r s ♣♦ss r♦ss t ♥tr♥ t① t♦r ♥ t é♥♦ t① ♥t♦♥ ②♠♣♦s♥ t s♠ rt♦♥ trts ♦♥ t ② ♦ r s♠♠r③ ♥ s ♥ ♦ ♠♥t♥ ♦s ♦♥ t t① ts ♦ ♦r ♠♦ sss t rt♦♥strt② ♦r ♥♦♥t① ♣♦② ♣r♠trs ♥ ♣♣♥① st ①♦♥♦s ♣r♠trs sr s♠♠r③ ♥ rt♦♥ ♦ t① ♣♦② ♣r♠trs s sr ♥ t♦♦♥ st♦♥

① P♦② Pr♠trs ♥ rts

❲ t ♦s♦s ♦ ♦r♥ s j = 1, . . . , R ♦♥ t t s tr♦ ♥ rtr ♦s♦s j = R+1, . . . , J ♦♥ t t s tr♦ s sss ♥ ♣♣♥① ♦♥strt ♣r♦tt② t②♣s z = 1, . . . , 5 t♦r♣rs♥t ♥♦t♦♥ ♦ t♠ ♦r ♥♦♠ q♥ts ♦r ♦t s♥ ♥ ♠rr ♦s♦s rs♣t② ❲ t t ♠② ♦♠♣♦st♦♥ f = m t♦ ♦♥ t ♠rr♦s♦s ♥ t①s ♦♥t② ♥ t f = s t♦ r♣rs♥t s♥ ♥ ♥♦♥♦♥t ♥♦s♦s ♥② ♦♥sr t ♦ ♥ ♠rr ♦s♦ ♥ ♦♥t② t♦♦rrs♣♦♥ t t ♦ t ♣r♠r② r

♦ ♠♣ t ♦s ♦ ♥♦♠ r ② ♦s♦s t♦ tr ♠♣r ♦♥tr♣rts ♦♥sr (i) r♦ss ♦r ♥♦♠ t♦ ♦rrs♣♦♥ t♦ t s♠ ♦ P♦♠♣r

♥♦♠ ♦♥♣t sr ♥ ♣♣♥① ♥ (1 − α − g) sr ♦ ♣sstr♦ s♥ss ♥♦♠ ♥ (ii) r♦ss ♣t ♥♦♠ t♦ ♦rrs♣♦♥ t♦ t s♠ ♦♥trst ♥♦♠ ♥s r③ ♣t ♥s ♥ ♥ (α + g) sr ♦ ♣sstr♦s♥ss ♥♦♠ ❲ ♦♥sr t s♥ss t① ts t♦ ♦rrs♣♦♥ t t♦s ② ♦r♣♦rt♦♥s

rt♦♥ rt♦s ♦r ♦r ♥ ♣t ♥♦♠ χi,z,f ♥ χa tr♥s♦r♠ ♦s♦s ♦♥♦♠ ♥♦♠ ♦ ♥t♦ t① s s s♣ ♥ qt♦♥ ❲♦ ♦r ♣r♦tt② t②♣ ♠② ♦♠♣♦st♦♥ s♣ rt♦♥ rt♦s ♦r ♦r ♥♦♠t♦ ♣tr t rt♦♥ ♥ ♥♦♥t① ♦r ♦♠♣♥st♦♥ r ② ♦s♦s ♦rt♠ ♥♦♠ q♥ts s r st ①♦♥♦s② s t rt♦ ♦ t① ♦r ♥♦♠t♦ P♦♠♣r ♥♦♠ s t ② t r s s♥ rt♦♥rt♦ ♦r ♣t ♥♦♠ ♦s ♦r t ♠♦ t♦ ♥rt t sr ♦r r♣ts r♦♠ t ♣t ♥♦♠ t① ♦r ♥ s♣t♦♥ ♦ t ♣t ♥♦♠r t① rt ♥t♦♥ ♥ qt♦♥ t s st ♥♦♥♦s② s♦ tt rt♣t ♥♦♠ t① r♣ts rt t♦ rt ♦t♣t ♥ t ♠♦s ♣rs♥t st②stt qr♠ r q t♦ s t ♦ t♦t ♦s♦ ♣t ♥♦♠t① ts s t ② t rt t♦ P ♥ s ♣r♦t ② t♦♥rss♦♥ t

r ♦r♥♠♥t ①t♦♥ ♦s♦ ♥♦♠

①t♦♥ ♦ ♣t ♥♦♠ ♥ ♦r tr♠♥t ♥ ♦s♦s ts♠ rt ♦ rtr♥ ♦♥ ♣♦sts ♥ rt♦♥ rt♦ ♣♣ t♦ ♣t ♥♦♠ ♥ ♦rr♥♦ ♦s♦s ② ♣t ssts s q♥t t♦ ♥ ♦rr♥ ♦ ♦s♦s ② t① ♣t♥♦♠ ❲ ①♣♦t ts q♥ ♥ s♣② ♥ r t① rt ♥t♦♥ ♦r ♣t♥♦♠ tt ♣♥s ♦♥ t rt ♦t♦♥ ♦ ♦s♦s sst ♦♥s ♥ t r♦ssst♦♥ strt♦♥ ♦ ssts ♦r ♦r♥ s♥ ♦r♥ ♠rr rtr s♥♥ rtr ♠rr ♦s♦s s ♥ qt♦♥ s ♥t♦♥s ♦ ♦s♦st① t② ♦♥ ♣t ♥♦♠ ♠② r② s♥♥t② ♦r tr ② t♦ ♥♣♥♥t ♦ tr ♣r♠♥♥t ♣r♦tt② t②♣ ❲ ss♠ tt ts ♥t♦♥ ts qrt ♦r♠

τat = q0fj + q1

fj (a

z,ft,j ) + q2

fj (a

z,ft,j )

2

s ♥t♦♥s r t t♦ t ♦sr ♣rs♥t r t① rts ♦rr ② t① ♣t ♥♦♠ s t ② t ♦r r♦♣ ♠② ♦♠♣♦st♦♥♠♦r♣ ♥t♦♥s r t♥ ♠♣♣ t♦ t ♠♦♥rt r♦ssst♦♥ strt♦♥ ♦ ♣t ssts ♦r ♠♦r♣ rstrt♥ t① rts t♦ ♥♦♥♥t r t① rt ♦♥ ♣t ♥♦♠ ♦r ♦s♦ s tr♠♥ ② t♥ ts♠♣♣♥ t tr rt ♦ ♣t sst ♦♥s ♥ ♠♦r♣ r♦♣ (f, j)

①s ss♦t t t rtr♠♥t s②st♠ ♥ ♦r ♠♦ ♥ t ♣♦rt♦♥♦ t ♣②r♦ t① ♥♥s t ♦ rt② s②st♠ s s s♣ t①trt♠♥t ♦ t ♥ts r ② rtrs ♣②r♦ t① ♣♣s t♦ ♦r♥♦♠ ♣ t♦ t trs♦ s♣ ♥r ♣rs♥t t rt ♦ rt♦♦r t t① s ♦ st r♦ss ♦r ♥♦♠ sts ♥♦♥♦s② ♥ ♥♦r♠②r♦ss t①♣②rs s♦ tt t♦t ♣②r♦ t① r♣ts rt t♦ ♦t♣t r ♦t 4.3% s

t ♥ ♦♥♦♠ t♦♦ t♦ ♣r ❲ ♥ ♣rt ♠♣♦②rs ♥ ♠♣♦②s r♠t ♣②♠♥t ♦ t ♣②r♦ t① t②

ss♠ t ♠♣♦② r♠ts t t② ♦r s♠♣t②

s ♣r♦t ② t ♦r ♦ rt② ♣tr t s♣ t① trt♠♥t ♦♦ rt② ♥♦♠ t r t① rt ♦♥ ♥ts ♥ qt♦♥ ♦r rtrs sst ♦r ♣r♦tt② t②♣ ♠② ♦♠♣♦st♦♥ ♠♦r♣ t♦ ♠t t ss♦t♥♦♠t r t① rts t ② t

t♦ ♥rt♥ s♣♥s ♦s♦s ♥ ♥t qsts ♥ ♦♦♥t ♦r t ♥♦ ♦sts ♥♥ ♠ ①♣♥ss Λ rt♦♥ ♦ tsssts ♦t t♦ ♦♥s♠♣t♦♥ srs r♠♥r t① t tr rt τ beqt ♥ t♥ strt ♥♦r♠② t♦ t ♥ ♣♦♣t♦♥ ♣r♠trsΛ ♥ τ beqt r ♦♥t② ♥ ♥♦♥♦s② tr♠♥ s♦ tt r trt① qstsr ② ♦s♦s r ♣♣r♦①♠t② 0.9% ♦ rt ♦t♣t stt ♥t t① r♥ s ♣♣r♦①♠t② 0.12% ♦ P s ♣r♦t ② t ♦r

①t♦♥ ♦ ♦r ♥♦♠ ♥tr♥ ① t♦r ❯s♥ t t② rtrstt ♥ t ♦♥strt t ♥tr♥ t① t♦r t♦ rt② ♠♦ t ♦♦♥♣r♦s♦♥s t sttt♦r② t① rt s ♦r ♦r♥r② ♥♦♠ st♥r t♦♥r♥ ♥♦♠ rt t① rt ♦♠ ♠♦rt ♥trst t♦♥ stt ♥ ♦♥♦♠ ss ♥ ♣r♦♣rt② t① t♦♥s rt ♥ t♦♥ ♥t ♥st♠♥t♥♦♠ ♥ r srt①s ♥ ♣♥♥t r rt ♥ ♦♥ s♦ ①♣t②♦♥t ♦r ♥② ♣s♥ ♥♦r ♦t r♦♥s ♦ t t① ♣r♦s♦♥s ♦ ssst♥t t♦♥ ♦ t t ♠r♥ t① rt r♦♠ t sttt♦r② ♠r♥ t①rt ♦r s♦♠ ♦s♦s ♦ ♦♥strt ♦s♦s ♦r♥r② ♥♦♠ t① s ♣♦rt♦♥ ♦ t① ♣t ♥♦♠ t♦ tr t① ♦r ♥♦♠ t ♣♦rt♦♥ ♦ t①♣t ♥♦♠ trt s ♦r♥r② s st t♦ ①♦♥♦s② rs ♦r r③ t①♣t ♥♦♠ s ♦♠♣t r♦♠ t ① ♠♥♠③t♦♥ ♦r s ss♠ t♥t t① t♦r s tt ♥② ♣r♦s♦♥ ♠② r t① ts s t♥♣♥♥ t♠③t♦♥ ♦ t♦♥s ♦r stt ♥ ♦ t①s ♦♠ ♠♦rt ♥trst♥ rt ♥ ♥ ♦r ♦ t st♥r t♦♥

♥♠r ♦ ♣♥♥ts ♠ ② ♦s♦ ♠ttrs ♦r rt♥ rts ❲♥② q②♥ ♣♥♥t s ♦♥t ♦r t r♥ ♥♦♠ t① rt t♦♥st t① rt ♣♣s t♦ ♣♥♥ts ♥r t ♦ ♥ t ♣♥♥t rrt ♣♣s t♦ ♣♥♥ts ♥r t ♦ ♦s♦ rs ♦r ♦ tstr ♥s ♦ ♣♥♥ts r t♥ r♦♠ t ♥ ①♦♥♦s② ss♦t t (j, z, f) ♠♦r♣ t♥ t ♠♦ ♦r s♦♠ t♦♥s t rq♥② strt♦♥♦r t ♥♠r ♦ q②♥ ♣♥♥ts t♥ ♣rtr ♦s♦ ♠♦r♣ ss♦ s rq♥② strt♦♥ s s♣② ♠♣♦rt♥t ♦r ♠♦♥ t st ♥rss ♥ t ♥♠r ♦ r♥ ♥♦♥♥r②

② ♥♦r♣♦rt♥ ♣♥♥ts ♥ t ♠♦ rt ♥♦tr s♦r ♦ tr♦♥t② s t ♣♦t♥t t♦ ♥♥ ♦♥♦♠ ♦s ♦r ①♠♣ t t ♥ttt ②♦♥ s♥ ♦s♦s r ♣♥♥ts ♦♥ r t♥ ②♦♥ ♠rr♦s♦s ♦t ②♦♥ s♥ ♥ ♠rr ♦s♦s ♦ ♥ t ♣s♥r♥ t t s♠ ♦ r♥♥s t ♠r♥ ♥ r t① rts r♦r t s♥ ♦s♦ t♥ ♦r t ♠rr ♦s♦ s t rr ♦rt ♠rr ♦s♦ ♥ ♠♦r q②♥ ♣♥♥ts t ♠♦♥t s t♥♥ ♦r ♦s♦s t ③r♦ ♣♥♥ts ♦r ①♠♣ t♥ t s♥ ♦s♦ t ♠♦r t♥ t ♠rr ♦s♦ r♥t ♥ t♦ ♠r♥ ♥r t① rts ♦ ♥ r♥t ♦r r♦ss ts t♦ ♦s♦s

s ♥♠r rts qsts tt r t t♦ r♣♥ts ♦tr t♥ s♣♦s ♦♥ r ♥ t ❯r♦♠ ❲♥ ♦r s♠♠r② ♦ ❯ qst t

♦ ♠♣♦s s♣♥ ♦♥ t t① ts ♥rt ② t ♥tr♥ t① t♦r s ♣r♦tt② t②♣ ♠② ♦♠♣♦st♦♥ s♣ tr♥srs traz,f ♦s♥ s♦ tt♥♦♠t r t① rts ♦♥ ♦r ♥♦♠ ♥rt ② t ♠♦ ♥ trt ♣rs♥t s♥ ♦r ♥ t tr♥t s♥r♦ ATRz,f ♠t t ♥♦♠

t r t① rt trts ♥rt ② t ATRz,f

♥♦♠tr t① rt ♦♥ ♦r ♥♦♠ ♥rt ② t ♥tr♥ t① t♦r ♦r ♥♦r♥ ♦s♦ ♦ ♣r♦tt② t②♣ ♠② ♦♠♣♦st♦♥ s

ATRz,ft =

∫

Jtaxz,ft,j Ω

z,ft,j dj

∫

Jiz,ft,j Ω

z,ft,j dj

=

∫

J

(

max

τ it iz,ft,j , 0

− crdz,ft,j − traz,ft

)

Ωz,ft,j dj

∫

Jiz,ft,j Ω

z,ft,j dj

r Ωz,fj s ♥ ♠sr ♦♥strt ♦r ts t♦♥ ♦♥② ts ♦s

♦s t ♣♦st ♦r ♥♦♠ tr♦r ♠♥t♥ ♥♠♣♦② ♥ rtr ♦s♦s

Ωz,fj =

Ωz,sj nj > 0 ♥ f = s

Ωz,mj n1

j > 0 ♥ f = m

0 ♦trs

tt♥ ATRz,f = ATRz,f

s♦♥ t ♦ qt♦♥ ♦r traz,f s s t sr ♦ ♣r♦tt② t②♣ ♠② ♦♠♣♦st♦♥ s♣ tr♥sr ♣②♠♥ts

traz,f =

∫

J

(

max

τ it iz,ft,j , 0

− crdz,ft,j − ATRz,fiz,fj

)

Ωz,fj dj

∫

JΩz,f

j dj

①t♦♥ ♦ ♦r ♥♦♠ é♥♦ ① ♥t♦♥ é♥♦ t① ♥t♦♥♣r♠trs λf1 , λ

f2 r rt s♦ tt t ♥♦♠t r ♥ t

♠r♥ t① rt ♦r ♠② ♦♠♣♦st♦♥ f ATRf♥ MTR

frs♣t② ♠t

t♦s ♥rt ♥ s♥ qr♠ ♥r t ♥ t s♣t♦♥ ♦ t ♥ qt♦♥ t ♥♦♠t r t① rt ♦r ♠② ♦♠♣♦st♦♥ f♥ ①♣rss s

ATRf =

∫

Z

∫

Jtaxz,ft,j Ω

z,ft,j dj dz

∫

Z

∫

Jiz,ft,j Ω

z,ft,j dj dz

=

∫

Z

∫

J

(

iz,fj − λf1 (iz,fj )1−λf

2

)

Ωz,fj dj dz

∫

Z

∫

Jiz,fj Ωz,f

j dj dz

r Ωz,fj s ♥ s ♥ qt♦♥ tt♥ ATRf = ATR

f♥ rrr♥♥ ♦r

λf1 ②s

λf1 = (1− ATRf)

(∫

Z

∫

J(iz,fj )Ωz,f

j dj dz∫

Z

∫

J(iz,fj )1−λf

2 Ωz,fj dj dz

)

♥♦♦s② t ♥♦♠t t ♠r♥ t① rt ♦r ♠② ♦♠♣♦st♦♥f ♥ ①♣rss s

♠ ssr♣ts r s♣♣rss ♦r st② stt t♦♥s

MTRf =

∫

Z

∫

J

(

∂taxz,ft,j /∂iz,ft,j

)

iz,ft,j Ωz,ft,j dj dz

∫

Z

∫

Jiz,ft,j Ω

z,ft,j dj dz

= 1− (1− λf2)(1− ATRf )

tt♥ MTRf = MTRf♥ ATRf = ATR

f t ♦ qt♦♥ ♥ s♦ ♦r λf2

♥ tr♠s ♦ ♦♥② ①♦♥♦s t① rt trts

λf2 =MTR

f− ATR

f

1− ATRf

♥ t ♥t ♣rs♥t s♥ st traz,f = 0 ♥r t t① s②st♠

r ♦♠♣rs♦♥ ♦ t ♥ t ♥ r s♦ t rt♦♥s♣t♥ st r♦ss ♦r ♥♦♠ ♥ t ♣♦rt♦♥ ♦ ♦s♦ t① ts ttrtt♦ ♦r ♥♦♠ ♦r ♦t t t♦♣ ♥ t ♦tt♦♠ ♥ t ♥t ♣rs♥tst② stt qr♠ s♠♦♦t② ♠♣s t① ♦r ♥♦♠ ♥t♦ tss♦t t① ts t t rt♦♥s♣ tr♠♥ ② t♦ ♣r♠trs ♦♥ t ♦tr ♥ s ♥♦♥s♠♦♦t ♠♣♣♥ tt ♦s ♦r s♦rs ♦ rt♦♥ ♥ ♦r♥♦♠ t① ts ♦tr t♥ t① ♦r ♥♦♠ ♣r♦s♦♥s ①♣t②♠♦ ♥r ts t① s②st♠ ♦ rt♦♥ ♥ ♦s♦s t① ♦r♥r② ♣t♥♦♠ ♥♠r ♥ ♦ ♣♥♥ts ♥ t①♣rrr ♦♥s♠♣t♦♥ ♦s t♦t t① ts ttrt t♦ ♦r

r♥s t♥ t t♦ t① s②st♠s r s♣② str t t ♦ ♥ ♥s♦ t ♥♦♠ strt♦♥ t t ♦ ♥ t① t② s ♦ ♦r s♦♠♦s♦s ♥r t t ♥r s ♦ ♥r t t t ♥♠rr ♦s♦s r♥♥ t s♠ ♠♦♥t ♦ ♦r ♥♦♠ ♦t t①t② r♥♥ r♦♠ t♦ ♥r t ① t② s ♣♣r♦①♠t②♥r ♥ st r♦ss ♦r ♥♦♠ t ts r♥ ♥r t

s sr ♥ ♣♣♥① t t♠ r ♦ t ♦r ♣r♦tt② tr♠s③z,fj r st s♦ tt r ♥♥ ♦r ♥♦♠ ♦r ♦r♥ s ♦r (z, f) ♠♦r♣ ♠ts trt s ♥ t s♥ st② stt qr♠ s♦sr ♥♥ st r♦ss ♦r ♥♦♠ s ♥ ♥ qt♦♥ ♦r ♦ t♦s♠♦r♣s s s ♦♥ r ♦r ♦s♦ ♠② ♦♠♣♦st♦♥ ♥r ♦tt ♥ t ♥ ss ♠♦♣r♦ ♦s② ♠ts t trt s♦s t ♦rrs♣♦♥♥ t① t② ♦r ♦ ts r ♥♦♠s ❲ t ss t tr♥sr ♣②♠♥ts traz,f t♦ ♠t ts trts ♦r ♠♦r♣ t s rt t♦ ♠t r t① ts t t ♦s♦♠② ♦♠♣♦st♦♥ ♦♥tss t ♠ts t rt trt t s♠r ♦ ♣rs♦♥ s t

r ♦r♥♠♥t ①t♦♥ ♦ s♥ss ♥♦♠

❲ ♠♦ ♦♠♥ s♥ss st♦r t♦t st♥s♥ t♥ ♦r♣♦rt ♥ ♣sstr♦ ♣r♦t♦♥ ♦r s♠♣t② ♥ ♦♥② ♦r♣♦rt♦♥s s♥ss t①ts ss♠ tt ♦ t ♦♠♥ s♥ss st♦r r♣rs♥ts ♦r♣♦rt♥tts ♦♥ t① t t rs r♣rs♥t♥ ♣sstr♦ ♥tts s r s t

s rs ♦ ♥♦t ♥ ♣②r♦ t①s ss♦t t ♦s♦ ♦r ♥♦♠

r ♦r♣♦rt sr ♦ t♦t ♥ ♣r♦t♦♥ ♦r s ♦♠♣tr♦♠ t r sr ♥ ♥♥ ♦♥ts

❲ t ♥r t① rt τ bust t♦ r♣rs♥t t r t ♠r♥ ♦r♣♦rtt① rt ♥ ①♦♥♦s② st t t♦ ♦ t rs♣t rt ♦♠♣t t♦ t s♦r♣♦rt ① ♦ ❲ t♥ st t r ♠♣s♠ t♦♥ lsdbust ♥♦♥♦s②s♦ tt r s♥ss t① ts rt t♦ rt ♦t♣t ♥ t ♠♦s ♣rs♥t st② stt qr♠ r q t♦ 1.19% s t ♣r♦t r ♦r♣♦rtt① ts rt t♦ P s ♣r♦t ② t ♦r

r ♦r♥♠♥t r♥sr P②♠♥ts

rtrs rr♥t ♦ rt② ♥ts ♣♥ ♦♥ tr ♣st ♦r ♦r♥♦♠ ♦♥ ts ①♣t ♣♥♥ ♥r t ss♠♣t♦♥ ♦ rt♦♥t② rqrs ♦s♦s t♦ ♦♥sr ♦qr♠ ♣ts t rs♣t t♦ s♦ srt② ♥ts♥ ♦r s♣♣② s♦♥s r t② ♠ ♥ ts ♣♣r♦ ♦ sst♥t♦♠♣tt♦♥ t♠ ss♠ tt ♦s♦s r ♦♥②rt♦♥ ♥ ts ♠♥s♦♥♥ ♦ ♥♦t ♦♥t♠♣t t ts ♦♥ tr tr s♦ srt② ♥ts ♥ ♠♥rr♥t ♦r s♣♣② s♦♥s t s ♦r s♣♣② ♦s ♥ ♥ ♣st ♦r ♦r ♥♦♠ r ♦♥sst♥t t t s♦ rr♥t srt② ♥ts ♦♥② ♦rt ♦♥qr♠ ♣t ♦r z, f) ♦s♦ ♠♦r♣ ♦♥ r ♦r ♥ ♦♦rt rtr♠♥t ♥ts r ss♠ t♦ ♥t♦♥ ♦ r t♠ r♥♥s♦r♥ t♦ t ♥t t♦r r♦♠ t ♦ rt② ♠♥strt♦♥

♣r♣♦s ♦ r tr♥srs trwz,ft,j ♥ ♦r ♠♦ s t♦ ♥sr tr ①sts s

s♦t♦♥ t r② ♦ s ♦ ♦♥s♠ rs♦rs ♥ t ♣rs♥ ♦ ♣♦st ♦r ♦♥s♦♥ ♦s♥ ♥ ♥♦♥♦s♥ ♦♥s♠♣t♦♥ s tr♥srs r st t♦ t ♠♥s tst♥ qt♦♥ ♦s ♦r ♦s♦ t♦t s♥s ♥ ♦s ♦r ♥♦♠s ♥♦t ♥♦ t♦ ♦r ♦♥s♠♣t♦♥ ♥ r♥t ♠♥♠♠s t♦ ♦♥s♠ ♥ r ♠♣s♠ tr♥sr trlt s ♥♦r♠ r♦ss ♦s♦s ♥ st ♥♦♥♦s② t♦ ♦t3.07% ♦ rt ♦t♣t rts t s ♣r♦t♦♥ ♦r t sr ♦ rr♥tr ♦r♥♠♥t tr♥srs ♥rt t♦ tr♥srs ♦r ♥ r s st ♦t② ♣♦rt♦♥ ♦ r♥ t① rts ♦r st② t ♠♣s♠ t①s lsttr st ♥♦♥♦s② t♦ ♦t 0.71% ♦ rt ♦t♣t rts t s ♣r♦t♦♥ ♦r ①s t① ts ♥ ♠s♥♦s ♣♥ts

tt ♥ ♦ ♦r♥♠♥t ①t♦♥

❲ ♥♦r♣♦rt ♦s♦ stt ♥ ♦ t① ts s♥ t ♥r t① rt τ slt♣♣ t♦ ♦s♦s st r♦ss ♦r ♥♦♠ ♥ t ♥r t① rt τ slpt ♣♣t ♦ ♦♥r♦♣ ♦s♥ ♦r♠r rt s ①♦♥♦s② st t♦ ♥ trt r♣rs♥t♥ t rtr ♦ stt ♥ ♦ t① ♥♦♠ ♦r ss t① ts ♦r t① ♥t s ♦♠♣t ② t ♦r ttr rt s ①♦♥♦s② st t♦0.0105× 0.7174 = 0.0075 s t ♣r♦t ♦ t ♥t♦♥ r ♣r♦♣rt② t① rt♦♠♣t s♥ stt st♠ts r♦♠ t t♦♥ ss♦t♦♥ ♦ ♦♠rs ♦r ♥ t t r ♣♦rt♦♥ ♦ t♦t rs♥t ♣t tt s ♥♦t ♦♥s♠r

♦r sr♣t♦♥ ♦ t ♦♥t ♦♠♠tt ♦♥ ①t♦♥s ♦r♣♦rt ♦ s ❲ ♥ ♣rt ♦r r♥♥s r♦♠ t st ②rs r s ♥ t ♥t

t♦♥ ♦r s♠♣t♦♥ ♣r♣♦ss ss♠ ♥ts ♣♥ ♦♥ t ②rs ♦ ♦r♥ ♦r♦s♦s tt♣sss♦♣s♣ ♦r sr♣t♦♥ ♦ t ♥t t♦♥

♥ ❯♣t t♦ t t ♥ ♦♥♦♠ t♦♦

rs s r♣♦rt ② P ♦r ttr tr♠ ♥ t ♣r♦t s ♥t♦ ♦♥t ♦r t tt ♦r ♥t♦♥ ♦ ♦s♥ srs ♥s ♦♥s♠r rs

♦r s♥ss t①t♦♥ t t stt ♥ ♦ st t ♥r t① rt τ slbt

♥♦♥♦s② s♦ tt stt ♥ ♦ s♥ss t① ts rt t♦ rt ♦t♣t♥ t ♠♦s ♣rs♥t st② stt qr♠ r q t♦ 0.38% s tst♦r r stt ♥ ♦ ♦r♣♦rt t① ts rt t♦ P s ♦♠♣tr♦♠ t t♦♥ ♥♦♠ ♥ Pr♦t ♦♥ts ♦r

P♦② ①♣r♠♥ts

❲ ♠♦♥strt t ♠♣t♦♥s ♦ ①♣t② ♠♦♥ t① ♣r♦s♦♥s ♣♣ t♦ ♦r♥♦♠ ② s♠t♥ t♦ ♣♦② ♥s ♥ tr♥ (i) t♥♣r♥t rt♦♥ ♥ sttt♦r②t① rts ♣♣ t♦ ♦r♥r② ♥♦♠ ♥ (ii) ♥ ①♣♥s♦♥ ♦ t r♥ ♥♦♠ t①rt ♦r ss ts ♦ ts ♥ ss♠ ♥ ♦ t s♠t♦♥s tt♥② t① ♣r♦s♦♥ st t♦ ①♣r tr r ♥st ♣r♠♥♥t ♥♥ t ♠♦r♣r♦s♦♥s ss♦t t P r st t♦ ①♣r ♥

t② tt ♥②ss ❲ ♦♠♣r t ♦♥♦♠ ♥ r r♥s r♦♠ t♥t ♣rs♥t st② stt t♦ t tr♥t st② stts r♦ss t① s②st♠♥s t♦ ② ♠r♦♦♥♦♠ rts r♦ss st② stts r r♣♦rt s ♣r♥tr♥s r♦♠ ♥t st② stt t♦ tr♥t st② stt ♥ ♠sr♥ ♥s t♦r ♦r ♦s♦s ♦♠♣t t ♣r♦♣♦rt♦♥ ♥ t♦ t t♠ ♣t ♦ t♦♥s♠♣t♦♥ ♦♠♣♦st xj ♦r ♦s♦s ♦r♥ ♥ t tr♥t st② stt tt ♦ ♥ t♦ ♠ t♠ ♥r♥t t♥ ♥ ♦r♥ ♥ tr st② stt ♦r s♥ ♦s♦ ts s

J∑

j=1

(βj−1πj)Uz,fj

(

(1 + ωz,f )xj , n

j

)

=J∑

j=1

(βj−1πj)Uz,fj

(

xj, n

j

)

r ωz,f s t ♣r♦♣♦rt♦♥ ♥ ♥ t ♦♥s♠♣t♦♥ ♦♠♣♦st rqr t♦ ♦t♥♥r♥ t♥ st② stts ♥ t ♥ s♣rsr♣ts ♥♦t qr♠♦ rs ss♦t t t tr♥t ♥ ♥t st② stts rs♣t② ♥t ♦ ωz,f ♠♣s tt ♦s♦ ♠♦r♣ (z, f) ♦ ♣rr rt ♥t ♥ st② stt rt t♦ t ♥t st② stt ♦ ♦♠♣t ♥s t♦ rtr ♥trt ♦t ss ♦ qt♦♥ ♦r ♦t s♥ ♥ ♠rr ♦s♦s♦r ♣r♦tt② t②♣ ♥

∫

Z

J∑

j=1

βj−1πj

U z,sj

(

(1 + ω)xj , n

j

)

Ωz,sj + U z,m

j

(

(1 + ω)xj , n1,j , n2,

j

)

Ωz,mj

dz

=

∫

Z

J∑

j=1

βj−1πj

U z,sj

(

xj, n

j

)

Ωz,sj + U z,m

j

(

xj, n1,, n2,

j

)

Ωz,mj

dz

♦r st ♦ ①♣r ♥ ①♣r♥ t① ♣r♦s♦♥s ♦r ②rs tr♦ ♥ t ♦♥s♠♣t♦♥ ♦♠♣♦st xj ♥sts ♦s♥ ♥ ♥♦♥♦♥s♠♣t♦♥ ♦♥s♠♣t♦♥ s ♥

♥tr② ♦♠♦♥♦s s♦♥ ♦♥ ♥ ♥tr♣rt ωz,f s t rqr ♣r♦♣♦rt♦♥ ♥ ♥ ♥strs s♠t♥♦s②

r ω s t rt ♣r♦♣♦rt♦♥ ♥ ♥ t ♦♥s♠♣t♦♥ ♦♠♣♦st ♥ t♦♠ t ♦s♦s ♥r♥t ♦♥ r t♦ ♥ ♦r♥ ♥ tr st② stt

♦r ♣r♣♦ss ♦ ♦r st②stt ♥②ss r t s ① ♥♦♥♦r♥♠♥t ♦♥s♠♣t♦♥ ①♣♥trs st t♦ ② ♥ t r t ♦ ♦ ts ♥str♠♥t ♦r ♠♣s♠ tr♥srs ♦s t ♥♦♠ t ♦♥ ♦s♦♦r ♦ tt ♦ ♦r ♠♣s♠ tr♥srs r s t♦ ♥ t t♥ ♦♥ r t ① r♦ss st② stts strt ② r♦♠ t rt♦♥♥ t ①t♥t ♦ r♦♥♦t r♦♥♥ tt ♠② ♦r ♥r t① s②st♠ t♦ r♥t ♣ts ♦ t ♠t♦♥ ♠t♦♥ s t♦♥st♥t ❵♦♠♣rt stt st② stts r ♥♦t t q♥t t♦ t st② stts r ♥r ttr♥st♦♥ ♣t ♥②ss s ♦

r♥st♦♥ ♥②ss ❲ ♥②③ t tr♥st♦♥ ♣t ♠♠t② ♦♦♥ t ♠♣♠♥tt♦♥ ♦ tr♥t t① ♣♦② ♥♥♥ r♦♠ ♥ ♥t st② stt ss♦tt ♣rs♥t t① ♥ ♦ ♦♠♣r rs♣♦♥ss r♦ss t① s②st♠s r♣♦rt ♥st♦ ② ♠r♦♦♥♦♠ rts ♦r t rst t♥ ②rs ♦♦♥ t ♣♦② ♥ t♦♦♥ t t ❵t ♥♦ s ② t ❯♥t tts ♦♥rss t♦ ♥♦r♠ st s♦♥♠♥ ♦r t① s②st♠ t srs r ①♣rss rt t♦ ♥tst② stt s

♦♦♥ t ♥♥♦♥♠♥t ♥ ♠♣♠♥tt♦♥ ♦ ♣♦② ♥ ♥ ❵②r 1 s ss♠ t♦ ♥♥t♣t ♥ t ❵②r 0 ♥t st② stt ♥ts ♥ t ♠♦ ♣rt ♦rst rr♥ t tr t♠ ♣t ♦ ♣♦② ♥ t ♦♥♦♠② ♥②♣♦②♥rt r t ts ♦r sr♣ss r ♥♥ ② ♦rr♦♥ ♦r st♦ ♣② ♦♥ ①st♥ t ♦r t rst ②rs ♦♦♥ ♣♦② ♥ ♦♥♦r♥♠♥t ♦♥s♠♣t♦♥ ①♣♥trs ♥ ♠♣s♠ tr♥srs ssq♥t② st ②q ♠♦♥ts t♦ ♠♥t♥ ♦♥r♥ s sst♥t② s ♣s ♥ ♦r ②rs tr♦ s♦ tt t r t s ♥ trtr s♥t ♥♠r ♦ t♠♣r♦s s ♦s♥ ♦r t tr♥st♦♥ ♣t s♦ tt t ♦♥♦♠② rs ♥ st② sttss♦t t tr♥t t① ♥ ♦ t t♦ ♥ ♦r rt♦♥ ♥rt tr♥st♦♥ ♥②ss t tr♥t st② stts r ♦♦♥ t tr♥st♦♥ssr r r ♥♦t ♦♠♣r t♦ t t♦♥st♥t st② stts sr ♦

① ♥str♠♥ts ♥ P♦② ♥s

♦t t① s②st♠s r rt ♦r ♣♦② ♥ ② ♦♥ ♦♥st♥t ♥♦♠ ♥♦ rs ss♦t t t ♥t st② stt ♣rs♥t qr♠ ♥ st♥ t① ♥str♠♥ts t♦ trt t t♦t ♦♥♥t♦♥ r♥ t ♦r st ② t ♥s t♦ ♦r ♥♦♠ t①t♦♥ ♥r t r ①♣t②♥♦r♣♦rt ♥ t t① t♦r s s♣ ② t ♣♦② ♥ ❯♥r t ♥s t♦ r ♥ t ♠r♥ t① rts ♣♣ t♦ ♦r ♥♦♠ r ♠ ②♣r♠tr③♥ t t① ♥t♦♥ t♦ ♠t t♦s ♥s ♣rt ② t ♦r f♠♦r♣ ♥ ♦t t① s②st♠s tr♥srs traz,f r ♥ t♦ trt t strt♦♥♦ t ♦♥♥t♦♥ r♥ t r♦ss (z, f) ♠♦r♣s s ♣rt ② t ♦r ♦t t① s②st♠s qt♦♥ s rst♠t t♦ ♠t t ♥s t♦ rt① rts ♣♣ t♦ ♣t ♥♦♠ ♣rt ② t ♦r t ♣t ♥♦♠ q♥t

♦♥♥t♦♥ r♥ t s t st♠t ♥ ♥ t① r♣ts r♦♠ t♦s ♣r♦t ♥r ♣rs♥t s♥ ♦rst ♦♥ ♦♥st♥t r♦ss ♥t♦♥ ♣r♦t ♦r ♠♦r ts

s♣ t♦ t (f, j) ♠♦r♣ t t ♥ s t♦ ♠t ♣rt ♥t♦ ♣t ♥♦♠ t① ts

♦ t ♦ t rt ♦r ♥♥ts ss♦t t ♥ ♣♦②♥ r ♣tr ♥ t① s②st♠ ♦♠♣r t ♥s t♦ rt r t①rts ♥ t ♠r♥ t① rts t♦ t ♥♦♦s ♥s ♣rt ② t ♥ r♣♦rt rr♦rs ♦r t ♥s t♦ ♥♦♠t r t ♠r♥t① rts ♣♣ t♦ ♦r ♥♦♠ ♦♥ ♦♥st♥t ♣rs♥t qr♠ ♦s♦♥♦♠ s ♥ ♦ rs s♠ rr♦rs ♥t tt ♦t t① s②st♠s♦♠♣r② r♣t t t① rt ♥s ♣rt ② t ♦♥sq♥t② ♦♥♥ tt t t① ♥str♠♥ts ♥ ♦t t① s②st♠s r rt t♦ ♣tr t rt ♦r ♥♥ts ss♦t t ♣♦② ♥

P♦② ①♣r♠♥t ttt♦r② ① t t♦♥

rst ♣♦② ①♣r♠♥t s ♣r♠♥♥t t♥ ♣r♥t rt♦♥ ♥ sttt♦r② t① rts♦r ♦r♥r② ♥♦♠ ♥ ♥♦♠ ♥trst ♥♦♠ s♦rttr♠ ♣t♥s ♥♦♥q ♥s ♥ ♣sstr♦ s♥ss ♥♦♠ ❲ s♠ ♣♦rt♦♥♦ t ♦♥♥t♦♥ r♥ t r♦♠ ts ♣♦② s t♦ t ♥ ♥ t① trt♠♥t ♦♣t ♥♦♠ ♥r② ♦ t rsts r♦♠ t ♥ ♥ t① trt♠♥t ♦ ♦r ♥♦♠

t② tt ♦♠♣rs♦♥

♦♠♣rs♦♥ ♦ st rts r♦ss st② stts s s♦♥ ♥ t t♦ t ♦♠♥s♦ ♦r ♦t t① s②st♠s ♥ s t ♣r♦tt②t ♦rs♣♣② ♥rss ♥ r♠s rs♣♦♥ t♦ t ♥rs ♠r♥ ♣r♦t ♦ ♣t ②♥rs♥ tr s ♦ t ♥ t ①♣♥ ♣r♦t♦♥ ♦ ♦t♣t ♥ s♠r ♠♥ts♥ t s♦rts r ♦♠♣t② ♥♥ ② rt♦♥ ♥ ♥♦♥ ♦r♥♠♥t♦♥s♠♣t♦♥ ♥ t tr♥t st② stt ♣rt ♣t s ♥♦t r♦♦t ② ♣t ♠t♦♥ ♥ ♥rs trt① rt rsts ♥ ♦s♦s ts♥t rs♦rs t♦ ♥rs tr ♦♥s♠♣t♦♥ ♦ ♠rt ♦♦s ♥ ♦s♥ srs

❲ s ♦ t ♦r s ♥♠♣♦rt♥t ♦r t ♣rtr ♥s t♦ rtssr ♦ t ♦ ♦ t① s②st♠ s ♠♣♦rt♥t ♦r t ♦r ♦ t①♣rrr♦♥s♠♣t♦♥ ♦s r♣♦rt t t ♦tt♦♠ ♦ ♥ t① t♦♥s r ♠♦ ①♣t② ♦♥② ♥r t t rt♦♥ ♥ sttt♦r② t① rts ♥ ts t① s②st♠♥trts t ts ♥r②♥ ♣r♦s♦♥s ♠♥s♥ t ♦ t♠③ t♦♥s♦r rt ♥ ♦♠♠♦rt ♥trst ♥ ♦ ♣r♦♣rt② t①s ♦♠ ♦s♦str♦r ssttt♦♥ t tt ts s s♥♥t ♦r rt② ♥ ♦♠♦♥rs♣ s t s ♥t ♥r r rt ♥ ♦♥r♦♣ ♦s♥♥ ♦♠♦♥rs♣ r r t ♥♦t ♥r t r r ♥rs t♦ t s♦ ♣rs♥ ♦ t ♥♦♠ t rt♦♥ ♥ t st♦ ♦ ♦♥r♦♣♦s♥ ♥r t ♦rs ♣rtr② r♦♠ ②♦♥ ♦s♦s ②♥ t ♣rs♦ ♦♠ s ♦♦s♥ t♦ r♥t ♠♥ts t t② ♦ ♦s♦ t♦ s ♦♥r♦♣♦s♥ s ♦tr ♦r ♦rr♦♥ ♥ s t①rr s♥ tr s rt②rr ♥rs ♥ t st♦ ♦ ♥♥ ssts ♥r t

♥r♦♥ ♥ P♥ ♦r sr② ♦ t ♣r ts ♦♥ rt ♥r ♥ r♥r ♥ tt t ♦♠♠♦rt ♥trst t♦♥s ♣r♦♠♦ts ♦♠♦♥rs♣

♦r ♥♦♠ ♦s♦s ♥ r♦♥s t ♥ st s♣♣② ♦ ♦s♥ rs ♥ ♦ ♥r♥s ♥ tt r♠♦ ♦ t t♦♥ s♦ rs t st♦ ♦ ♦♥r♦♣ ♦s♥

♦ ♦ t① s②st♠ s s♦ ♠♣♦rt♥t ♦r ♥s t♦ ♣r♦tt②♥t♠rt ♦r ♦rs s♣t s♠r② ♥rs t ♦r s♣♣② ♦rs st②♥r t t② ♥rs ♥r t s rst ♥ts tt t ♥t♠♣♦②♠♥t ♥ ♥r t s ♦♠♣♦s ♦ rt② ♠♦r ♦rs r♦♠ ♣r♦tt② ♦rrs ♥ rt② ss ♦rs r♦♠ ♦♣r♦tt② ♦rrs rst ♥t ♠♣♦②♠♥t ♥rs ♥r t s rt② ♥ r♦ss ♣r♦tt②t②♣s s r♥ strt ♥ r s r② t♦ t ♦r ♦s ♦ s♥ ♦rrs ❯♥r t rt s♥ ♦rrs ♦ ♣r♦tt② t②♣s t♦ tr♦♦r ♦♦s s♠r② r ♥s r♦♠ ♣rtt♠ ♦r ♥t♦ ♦t t♠ ♦r ♥♥♠♣♦②♠♥t t ttr ♦ ♦rs s r② rtr♠♥t ❲ tr s ♦♠♣r♠♦♠♥t ♥t♦ t♠ ♦r ♥r t t tr s ♥♦ ♣♦②♥ r② rtr♠♥t t♦♥ ♠♣♦rt♥t r♥s r ♦sr ♠♦♥ s♥ ♦rrs ♦ t ♦rt♣r♦tt② t②♣ ♥ ♠rr ♦rrs ♦ t rst ♣r♦tt② t②♣ r ♦rrs ♦♥ ♠♣♦②♠♥t stts ♦♦s ss ♦rs ♣r♦r t♦ rtr♠♥t ♥r t ♥t♦ ♥trt♦♥ t♥ t①♣rrr ♦♥s♠♣t♦♥ ♥ ♦r ♦rs ts ♦s♦ r♠♦♥ t♦s ♦ ♣♦②♥ ② ♥ ♦♠ ♣rs ♥ ts ♠t rt② rr st♦ ♦ ♥♥ ssts r♦♠ t♦ r ♦♥ ♣♦♥ ♦r♥♦♥♦s♥ ♦♥s♠♣t♦♥

r♣♦rts ♣♦②♥ r ♥s ♥ tr♠s ♦ t ♣r♥t ♥ ♥ t♠ ♦♥s♠♣t♦♥ ♥ t tr♥t st② stt tt ♦ ♥ t♦ ♠ ♦s♦s♥r♥t t♦ ♥ ♦r♥ ♥ t ♥t st② stt t t rt ♦t t ♥ t s♦ s♠r r ♠♣r♦♠♥ts t rs♣t ♦♥s♠♣t♦♥ q♥trt♦♥ ❱ ♦ −0.7% ♥ −0.8% ❲ s♠r r♥s ♥ ❱ ② ♣r♦tt②t②♣ ♦ s♦ s♦♠ rt♦♥ r♦ss t① s②st♠s t♦ ①♣t t① t ♥♦r♣♦rt♥r t ♥ ♣rtr t ♥ r ♥ ♠♦♥ t ♦st t♦ ♦r♣r♦tt② t②♣ s♥ ♦s♦s ♥r t rts tt t ①♣t ♠♦♥ ♦t st♥r t♦♥ s ts ♦s♦s t tt ♦r ♥♦ t① ♥♦♠ r♥r♥ t♠ rt② ♥t ② ts ♣♦② ♥ ♠r② t r♥ ♥ r♠♣r♦♠♥t ♠♦♥ t ♦rt ♣r♦tt② t②♣ s♥ ♦s♦ rts ♥trt♦♥ t♥ t r ♦ t♦♥s ♥ ♦r s♣♣② ♦r ♥r t t s t♦ ♣tr

r♥st♦♥ ♦♠♣rs♦♥

♦♥♦♠ tt② ♦r t rst ♦ t t♥♥ rtt ♥ ♦srr♦ss t① s②st♠s ♥ r ♦♦♥ t ♥t② rr ♥rs ♥ t ♦rs♣♣② ♥r t ♥ ♣rt t♦ rt② str♦♥ rs♣♦♥s r♦♠ ♣r♦tt②♦rrs r♠s ♦sr r ♠r♥ ♣r♦t ♦ ♣t ♥ rs♣♦♥ ② ♥rs♥tr s ♦ ♣t ♥ ♣r♦♥ ♠♦r ♦t♣t ♠t♦♥ ♦ ♣t s ♥♥② ♥ ♥rs ♥ ♦s♦s ♥♥ ssts t t ①♣♥s ♦ ♠rt ♥ ♦s♥♦♥s♠♣t♦♥ ❲ t st♦ ♦ ♦s♥ sr ♦♥s♠♣t♦♥ ♦♦s s♠r ♣ttr♥r♦ss t① s②st♠s t rt② rr rs ♥ ♦♥r♦♣ ♦s♥ ♥r t rts ts t① s②st♠ ♣tr♥ t r ♦ t♦♥s ss♦s♦s t♦ ♦ rt② ♠♦r ♥♥ ssts s ts rsts ♥ rt② rrt① ♥♦♠ ss ♥r t t r t① r♥ ♦ss s rt② s♠r ♥tr s ss t ♠t♦♥ ♦r t rst

r t tr♥st♦♥ ♣t t ♥rs ♥ t ♦r s♣♣② ♥s ♦r♦rs r♠♥ ♥r t ♥t♥ rt② rr ♦♥♥trt♦♥ ♦ ss

♣r♦t ♦rrs ♦r t♠ ♥r ts t① s②st♠ s ♦rs s t ♥t ♥ssq♥t ♥rs ♥ t trt① rt r rt② rr ♥r t t♦t rs♣t ♠♥t ♦ ♥s t♦ t ♦r ♥ s♥ss ♣t ♦r t rst ❲ ts ♣ttr♥ ♥s ♣r♦tt② ♦rrs t ♦♥ ♦rs r♦♠ rt② str♦♥r ♥♦♠ t t ♥s ♥ ♦♣r♦tt② ♦rrs t♦ ♥rs♦r ♦rs r♦♠ rt② str♦♥r ssttt♦♥ t

r♥♦rr r♥s ♥ rt rs♣♦♥ss r♦ss t① s②st♠s ♦r t rst sst♥t q♥ttt ♠♣t♦♥s ♦r ①♠♣ t♦ ②rs tr t ♣♦② ♥ t ♣r♦ts tt ♥♦♠♥ P ♦♥ r t♥ tt♣r♦t ② t ts ♣r♦t r♥ r♦s t♦ ♦♥ ② t t ②r♠r② t r♥t ♥ ♦r ♦rs ♠♦♥t t♦ ♦♥ ♠♦r ♣♦t♥t ♦rs ♥rt ♦♥ r ②r ♦r t rst ♥ t s s t ♣r♦tstt r r♥s ♦♥ r t♥ ♣r♦t ② ♦r rst s ♠♣s tt t ♣r♦t ♦ r t ② t ♣ t t ♥ ♦ t②r ♥♦ s ♦♥ ♦r ♥r t

P♦② ①♣r♠♥t ①♣♥s♦♥ ♦

❲ s♠t ♣r♠♥♥t ①♣♥s♦♥ ♦ t tt ♥rss t ♠①♠♠ rt♥ ♥s t ♣s♥ ♥ ♣s♦t r♦♥s ♦r ♦♥♦♠ ss ♦s♦s ♥♣rtr t ♠①♠♠ rt s ♥rs r♦♠ ts ♦ t♦ ♦r♦t s♥ ♥ ♠rr ss ♦s♦s rt r♠♥s ② r♥s ♦♠♣t② ♣s♥ ♦r ♦s♦s ♦ tr ♠② ♦♠♣♦st♦♥ t ♦r ♥♦♠ ♦ t ♥s t♦ ♣s ♦t t ♦r ♥♦♠ s ♦ ♥ ♦rs♥ ♥ ♠rr ♦s♦s rs♣t② ♦♠♥ ♦♠♣t② ♣s♦t t ♥ ①♣♥ s s s♦♥ r♣② ♥ r ①t♣♦s ♥stt ♣rs♥t s ♦r ss ♦s♦s ♥ ♦s♦s t ♦♥

♥ ♥s ♦♥t ♦r ♠♦st ♦ t ♦♥♥t♦♥ r♥ t ♦t st② r ♠♦r ② t♦ ss t♥ ♠rr ♦♣s ♥ s t② t♥t ♥♦♠ r♥ t♦ q② ♦r t ①♣♥ rt ❯♥r t ♣rs♥t s t ♦st♣r♦tt② s♥ ♦rrs r t ♣r♠r② rt r♣♥ts ❯♥r ttr♥t s ♦♥ ♦♥st♥t ♥t ♦r ♥♦♠ t ♦st tr ♣r♦tt② t②♣s ♦ r② q② ♦r t rt t t ♦rt ♣r♦tt② t②♣ q②♥ ♦♥② t ②♦♥r ♥ ♦r s t♦ ② ♣ttr♥s ♦ ♣♥♥ts ♥ ♦r♣r♦tt② ❲ t ♦st ♣r♦tt② s♥s ♦ ♦t ♦♥ t ♣s♥ r♦♥ ♦ t tr♥t s t s♦♥ tr♦ ♦rt ♣r♦tt② t②♣s♦ ♦♥ t t ♦r ♣s♦t r♦♥ ♦ t s ♥ rtr r♦♠ t ♠①♠♠rt r♦♥ s tr ♥♦♠ rss s ♥t ♣♦st♦♥♥ r♦♥ t tr♥ts rts ♥ ♥♥t ♦r ts ♥s t♦ ♥ tr ♦r ♦rs ♥ ♥♦r ♥♦♠ t♦rs t ♠①♠♠ rt r♦♥

s rs r ♦♠♣t rt t♦ t ♦♥rss♦♥ t s s♥ ♦rst ♣rs♥t♥ t ♥ ♦♥♦♠ t♦♦ t♦ ♥r t ss♠♣t♦♥ tt t ♣t ♦ ♥t♦♥ s♥t ② t ♣♦② ♥

t s ♥ ♦♠♥t tt t r♥ ♥♦♠ ① rt s ss♦t t sst♥t ♦r♠s ♥ ♥♦♠♣t t♣ Pr ♥ ♦r tr♠s ♦stt♥ ts ♦♥ t♦trt ♦t②s ♦♥tss ss♠ ♦♠♣♥ ♥ t♣ ♥ ♦r s♠t♦♥s s♦ tt t ♥ t① s②st♠s ♥ ♥②③ ♥ ♦♠♣r s♦♥

t② tt ♦♠♣rs♦♥

♥s t♦ st rts r♦ss st② stts r r♣♦rt ♥ t rt ♦♠♥ ♦ ♦r ♦t t① s②st♠s s t t rt rt♦♥ s♠t♦♥ t① s②st♠♣r♦s s♠r ♥s t♦ ♠r♦♦♥♦♠ tt② ♥ t tr♥t st② stt ♥ s ♦t♣t s♥ss ♣t ♥ t ♦r s♣♣② ♠r♥② rs♦r s♠r② s♠ rss ♥ t ♦r s♣♣② ♦♥♥ t r r♥s ♥ ♠rt ♦r ♦rs ♠♣② sst♥t rt♦♥ ♥ ♠♦r♣ rs♣♦♥ss r♦sst① s②st♠s ❯♥r t t ♥rs ♥ ♦r ♦rs ♦ s ♥t ♦ sst♥t ♥rs ♥ ♦r ♦rs ② ♦♣r♦tt② ♦rrs ♦t♥ rs ♥ ♦r♦rs ♦ ♠♣r♦tt② ♦rrs t♦ q② ♦r t ①♣♥ rt s ♦♥trstst t rs♣♦♥s ♥r t r t ♥t ♥ ♥ ♦r ♦rs ♦ ♣♣r♦①♠t②③r♦ rts r♦② q ♥rs ♥ ♦rs ② ♦♣r♦tt② ♥s ♥ rs ♥ ♦rs ♦ ♠♣r♦tt② ♥s s ♥t ♥t ♥ ♥ ♦rss ♥♦♥sst♥t t t ♥r ♥♥s ♦ t trtr ssts ♣♦st ♥t♦rs rs♣♦♥s ♥ ♦♥sr♥ ♠r♥s ♦ ♦r st♠♥t s♠t♥♦s② ❯♥r ♦t t① s②st♠s t s♠ ♥rs ♥ ♦s♥ ♥ ♠rt ♦♥s♠♣t♦♥ s r♥② ♦♣r♦tt② s♥ ♦rrs ♦ r ♥r② ♦r♥ ♠♦r rt② rr♥rs ♥ ♦♥s♠♣t♦♥ ♥ rs ♥ r t① r♥ ♥r ♦r rts r ♥rs ♥ ♦r ♦rs ♦ ts ♦rrs ♦ t ♣ t ①♣♥ rt

♠♣♦rt♥t ♠♣t♦♥ ♦ t t① s②st♠ ♦ ♦r ♦r ♦rs s s♦♥ ♥ r r ♦♠♣♦s t ♠♣♦②♠♥t rs♣♦♥s ♠♦♥ s♥ ♦rrs r♦ss ♣r♦tt② t②♣s ♦r t t♦♣ ♥ t ♦tt♦♠ ♦♥sst♥t t ss ♥ ♠♥ ♥ tt ♦r ♦t t① s②st♠s s♥ ♦rrs ①t ①t♥s ♠r♥ ♠♦♠♥t ♥t♦ t ♦r ♦r ♦♥ t ♥t♥s ♠r♥ ♦r ♥ tt ♥r t ♥t ♠♦♠♥t ♦t ♦ t♠ ♦r ♦♠♥ts ♥t ♠♦♠♥t ♥t♦ t♠ ♦r♥ ♦rs ♠♦♥ s♥s ♣♣r♦①♠t② ♥♥ ♥ ♦♠♥ t t ①t♥s♠r♥ rs♣♦♥s ♥st ①ts rs♣♦♥s r t ♠♦♠♥t ♥t♦ t♠♦r ♦♠♥ts ♦♥ t ♥t♥s ♠r♥ ♥rs♥ ♦rs ♦r ♥ ♠♥♥r ♦♥sst♥t t t ♥♥s ♦ tt② t s♣② ♠♦♥ t ♦st ♣r♦tt②♦rrs ❲ ♣r♦tt② s♥s s♦ ♠♦st ♥♦ ♥ ♥ ♠♣♦②♠♥t ♥rt tr ♠♣♦②♠♥t rs♣♦♥s s sst♥t ♥r t ♥ ts r♦♣ r♥s♦r ♥♦♠ ♦ t rt t② trs♦ ts rs♣♦♥s s t♦ t ♥t②♦ t t♦ trt t s♣ ♦s♦s rt② t ② ts ♣♦② ♥

❲ r♣♦rt t ❱ ♥ r♦ss st② stts t t rt t t① s②st♠ ①ts ♥ r r ♠♣r♦♠♥t s♠r t♥ t t① s②st♠t ❱ ♦ ♥ rs♣t② ♦♥sst♥t t t ♥♦t♦♥ tt ts ♣rtr♣♦② ♥ s ♠♦st ♠rr ♦s♦s ♥t rt② ♦sr tt ♠rr♦s♦s r ♥s tt r sst♥t② ♦ tt ♦ s♥ ♦s♦s st rst ♥rs ♦ t ①♣♥ rt r t ♦st t♦ ♣r♦tt②t②♣ s♥s♦r ♦t t① s②st♠s ♦sr r r ♠♣r♦♠♥ts ♠♦♥ ts ♠♦r♣ r♥ ♠♦♥ t tr ♣r♦tt②t②♣ s♥ ♦s♦s s t ♣r♠r② s♦r♦ rt ❱ r♥s s t r ♠♣r♦♠♥t ♦r t s sst♥t②rr t♥ tt ♦ t ♥ t ♠sr ♦ tr♣r♦tt② s♥s ♦ ♥t ♥♦♠ s t♥ t t② trs♦ s s♠ ts rst rts t ♥t②♦ t t♦ ♣tr t tr♥t ♥tr ♦ ts ♣♦② t s♠♦♦t ♣♣r♦①♠t♦♥t t♦ t t rst s ♥ ♦rstt♠♥t ♦ r ♠♣r♦♠♥t s♥t②

♦s ♥ ♦tst♥ ♦r r♥t sr② ♦ t trtr ♦♥ ♦r s♣♣② ♥ t

str♦♥ t♦ t t ❱ t t rt

r♥st♦♥ ♦♠♣rs♦♥

rt ♦t♣t ♥rss ♦r t rst ②rs ♦♦♥ t t♥♥ ①♣♥s♦♥♦ t r♥ ♥♦♠ t① rt s s♦♥ ♥ r ①t♥t t♦ t s r♥ ②t ♦r s♣♣② ♦r ♦ ♦s♦s rs qtt② ♥ q♥ttt② r♦ss t①s②st♠s ♦s♦s rs♣♦♥ t♦ t rt ①♣♥s♦♥ ② ♥rs♥ ♦r ♦rs ② ♥ ♦♥ r ♦r t rst ②rs ♥r t ♥ rs♣t② srt♦♥ r♦ss t① s②st♠s s ♥r② ♦t♦♠♦s s t ♠♣r ♥ sr② ②♦s ♥ ♦tst♥ ssts tt t s ♥♦♥♥ ♣♦st t♦♥ ♦r ♠♣♦②♠♥t s ①t ♥r t rt② rr t① ss ttrst ♥r t ♥rt rt② s♠r r r♥ ♦ss ♥ ♠t♦♥♦ ♣ t ♦r t rst ♦ t

r t s♦♥ ♦ t rst tr s st ♥ r♥ ♦rr ♦ rtrs♣♦♥ss r♦ss t① s②st♠s s♣t t ♥rs ♥ ♠♣♦②♠♥t r♦♠ t ♦st ♣r♦tt② ♠ ♣r♦tt② ♦rrs r ♦rs t♦ r t ①♣♥ rt ♥t t① s sr♥s s rt ♣②♠♥ts ♥rs ♥rs♥ t ♠t♦♥ ♦ ♣t ♦r t s♦♥ ♦ t ♠t ts ♥ t♦ r♦ ♦t ♣rt♣t ♦r♥② ♥r t① s②st♠ ♦r t s♦♥ ②rs r♠s ♥ ♦t t①s②st♠s rs♣♦♥ t♦ t rs♥ ♦st ♦ ♣t ② s♥ ss ♥ ♣r♦t♦♥ ♦r ts ♣r♦

♠♣♦rt♥ ♦ s♥ ♥ ①♣t t① s②st♠ ♦r ♠♦♥ trt ♣♦② ♥ss s ts s rtr rt ♥ t q♥ttt ♣r♦t♦♥s ♦r t ②rs ♦♦♥ ♠♣♠♥tt♦♥ ❲ ♥ ♦♥ ♠♦r ♣♦t♥t ♦r ♦rs ♦♥ r ②r ♦r trst ♥r t t♥ t ❲ ts s ttrt t♦ ♦st ♣r♦tt②♦rrs ♥rs♥ ♠♣♦②♠♥t ♠♦r ♥r t tr r ♦tr ♥♥ts rt♦r ♠ ♣r♦tt② ♦rrs s t r♦ss rt s ♦r t♠ s♦♠ ♦rrsrs ♠♣♦②♠♥t t♦ r t rt ♦r rr rt ♥ ♣②♦ts ♥rss rst ♦ ♠♦r ♦rrs st♥ ♦r ♦rs t♦ t ♥t ♦ t ①♣♥ t r t① r♥ r♥t t♥ t t♦ s②st♠s r♦s t♦ ♦♥♦r t rst ♠♣ ♦ r t ② t ♣ ♣r♦t ②t s ♦♥ rr t♥ ♥r t ② t ♥ ♦ t rst

♦♥s♦♥

s ♣♣r s ①♠♥ t ①t♥t t♦ ♦ ①♣t t① t ♥ tr♦♥♦s♥t ♠♦s s ♠r♦♦♥♦♠ ♠♣t♦♥s r♥t ♦r t① ♣♦② ♥②ss ❲ ♥♦r♣♦rt t♥ rs ♠♦ ♥ ♥tr♥ t① t♦r tt ①♣t②♠♦s ② ♣r♦s♦♥s ♥ t ♥tr♥ ♥ ♦ ♥ ♦♥t♦♥s ♦♥ ♦s②♥rt♦s♦ rtrsts ♥ tr♠♥♥ ♦r ♥♦♠ t① ts ❯s♥ ts t①s②st♠ ♥ ♥ ♥♦♥t♦♥ s♠♦♦t t① ♥t♦♥ ♥ tr♥ t♦ s♠t ♣♦② ♥s ♠♦♥strt tt t ①♣t ♠♦♥ ♦ t① ♣♦② ♥s ♦s ♦s♦s t♦st tr ♦r ♥ rt ♦♣t♠② t♦ t ♥ ♦♥t♥ ♦r ♥trt♦♥s♠♦♥ ♥r②♥ ♣r♦s♦♥s t♥ t t① ♦ ♥ ♣rtr ♦r t sttt♦r② t① rtrt♦♥ ♥②③ r ♦s♦s st tr t①♣rrr ♦♥s♠♣t♦♥ ♦r ♥rs♣♦♥s t♦ t ♥rt② ♠♥s ♦ t♦♥s ♦rr♥ t♦ ♥trt♦♥ tt rt rt♦♥ ❲ s♦ ♠♦♥strt tt t① t ♥♦r♣♦rt ♥ t ♥tr♥t① t♦r ttr ♣trs trt ♣♦② ♣r♦♣♦ss ♦r t ①♣♥s♦♥ ♦♥②

♦s♦s t♥ t ①♣♥ rt s st ♦rs t♦ r t rr rttr ♦ ts ♦r rs♣♦♥ss r ♦sr ♥ s♠t♦♥s r s♠♦♦t t①♥t♦♥ s s ♥ ♦ t ♥tr♥ t① t♦r

♥ t t♦♥st♥t st② stt ♥②ss t ♣♦②♥ ♦r rs♣♦♥ssrst♥ r♦♠ t ①♣t t① ♣♦② ♠♦♥ ♥r t ♥tr♥ t① t♦r r♥♠♣♦rt♥t ♦r ♥s t♦ ♠r♦♦♥♦♠ rts ♦♥ t♥♥ tr♥st♦♥♣t ♠♠t② ♦♦♥ ♦t ♣♦② ♥s ♦r ts ♦r s ss♦tt r♥t t♠ ♣t ♦ ♦r s♣♣② ♣rt ♣t ♥ ♣ t ♠t♦♥♥rt♥ q♥ttt ♥ qtt r♥s ♥ ♠r♦♦♥♦♠ rts r♦sst① s②st♠s s ♥♥s ♥t tt t s ♦ ♥♦♥t♦♥ s♠♦♦t t①♥t♦♥s ♠② ♣♣r♦♣rt ♦r st②stt ♥②ss t♥ tr♦♥♦s♥t ♠♦s t ♥s♦♥ ♦ ①♣t t① t s ♠♣♦rt♥t ♦r r tr♥st♦♥ ♥②ss ♦t① ♣♦② ♥s r♦r ♦♦s♥ t♦ s ♥♦♥t♦♥ s♠♦♦t t① ♥t♦♥s s ♣rs♠♦♥♦s s♦rtt ♦r ♠♦♥ ♥ ♥tr t① s②st♠ s ♠♥♥ ♦♥sq♥s♦r t ♣r♦t♦♥s ♦ ♦♥♦♠ tt② ♦♦♥ t① ♣♦② ♥s

s ♥ rs

♥♥ ♥tr♠r② ♥ t ♥ ♥♦♠ tt♠♥t

ssts ts ♥♦♠ ①♣♥ss

Dt+1 rptDt

Kt+1 rtKt δKKt + Ξt

Hrt prtH

rt δrHr

t−1

Bt+1 ρtBt

r st r♦ss ♦r ♥♦♠ ♥ t♦s♥s ♦

rt rt

Pr♦tt② ♥ ♦s♦s rr ♦s♦s

1

2

3

4

5

rt rt

♥ ♦s♦s rr ♦s♦s

rt

r ♦r ♥♦♠ ① t② ♥ t♦s♥s ♦

rt rt

Pr♦tt② ♥ ♦s♦s rr ♦s♦s

1

2

3

4

5

rt rt

♥ ♦s♦s rr ♦s♦s

rt

s♦t rr♦rs ♦r rt ① ts ♥s ♦♥ ♦r ♥♦♠

t t♦♥ ①♣♥s♦♥

r ① t rr♦r

t r♥ ① t rr♦r

rr♦rs ①♣rss s s♦t ♣r♥t ♣♦♥t r♥s r♦♠ rt ♥

t② tt ♦♠♣rs♦♥

t t♦♥ ①♣♥s♦♥

t♣t

s♥ss ♣t t♦

t ♦r ♣♣②

♦r ♦rs

rt ♦♥s♠♣t♦♥

♦s♥ r ♦♥s♠♣t♦♥

r ① ♥

♦s♦ ♥♥ ♣♦sts

♥r♣ ♦s♥ t♦

♦♠♦♥rs♣ t♦ ♣♣

rt ♥

rs t t ①♣t♦♥ ♦ t ♦♠♦♥rs♣ rt♦ r ①♣rss s

♣r♥t r♥s r♦ss st② stts

❱ ♦r t t♦♥

Pr♦tt② ♥ ♦s♦s rr ♦s♦s

1

2

3

4

5

rt

❱ ♦r ①♣♥s♦♥

Pr♦tt② ♥ ♦s♦s rr ♦s♦s

1

2

3

4

5

rt

r ♦r ♥♦♠ ♥ ① ts ♥tr♥ ① t♦r s ① ♥t♦♥

0 50 100 150 200

Adjusted Gross Labor Income (Thousands)

-5

0

5

10

15

20

Labor

Incom

e T

ax L

iabili

ty (

Thousands)

Tax Liability Under the ITC

Single

Married

300 350 400 450

Adjusted Gross Labor Income (Thousands)

35

40

45

50

55

60

65

70

75

80

85

Labor

Incom

e T

ax L

iabili

ty (

Thousands)

Tax Liability Under the ITC - High Income

0 50 100 150 200

Adjusted Gross Labor Income (Thousands)

-5

0

5

10

15

20

Labor

Incom

e T

ax L

iabili

ty (

Thousands)

Tax Liability Under the BTF

300 350 400 450

Adjusted Gross Labor Income (Thousands)

35

40

45

50

55

60

65

70

75

80

85

Labor

Incom

e T

ax L

iabili

ty (

Thousands)

Tax Liability Under the BTF - High Income

r t t♦♥ Pr♥t P♦♥t ♥ ♥ ♠♣♦②♠♥t tts ② Pr♦

tt② ②♣ ♥ ♠② ♦♠♣♦st♦♥ r♦ss t② tts

r t t rts r♥ r♥st♦♥ t t♦ ♥t t② tt

0 2 4 6 8 10

Time

1

1.02

1.04

Ratio

Output

0 2 4 6 8 10

Time

0.98

1

1.02

Ratio

Business Capital

0 2 4 6 8 10

Time

1

1.02

1.04

Ratio

Effective Labor

0 2 4 6 8 10

Time

1

1.02

1.04Ratio

Hours

0 2 4 6 8 10

Time

0.98

1

1.02

Ratio

Market Consumption

0 2 4 6 8 10

Time

0.95

1

1.05

Ratio

Housing Services

0 2 4 6 8 10

Time

0.95

1

Ratio

Tax Revenue

0 2 4 6 8 10

Time

1

1.1

1.2

Ratio

Public Debt

0 2 4 6 8 10

Time

0.9

0.95

1

Ratio

Owner-Occupied Housing

0 2 4 6 8 10

Time

1

1.05

Ratio

Financial Assets

ITC

BTF

r P♦② ①♣r♠♥t ①♣♥s♦♥

r ①♣♥s♦♥ Pr♥t P♦♥t ♥ ♥ ♠♣♦②♠♥t tts ② Pr♦

tt② ②♣ ♦r ♥ ♦s♦s r♦ss t② tts

r ①♣♥s♦♥ rts r♥ r♥st♦♥ t t♦ ♥t t② tt

0 2 4 6 8 10

Time

0.995

1

1.005

1.01

Ratio

Output

0 2 4 6 8 10

Time

0.99

0.995

1

1.005

1.01

Ratio

Capital

0 2 4 6 8 10

Time

0.996

0.998

1

1.002

1.004

Ratio

Labor

0 2 4 6 8 10

Time

0.99

1

1.01

1.02

Ratio

Hours

0 2 4 6 8 10

Time

0.97

0.98

0.99

1

1.01

Ratio

Consumption

0 2 4 6 8 10

Time

0.96

0.98

1

1.02

1.04

Ratio

Housing

0 2 4 6 8 10

Time

0.98

0.99

1

1.01

Ratio

Tax Revenue

0 2 4 6 8 10

Time

1

1.01

1.02

1.03

1.04

Ratio

Public Debt

ITC

BTF

r♥s

st♠♥♥ ♥ t♥s rr ♦sts ♦ r♥ ♦r♥ ♦ P♦t

♦♥♦♠②

♥ t r ♥ ♥♦♠ t① ♥t♦♥ s♣t♦♥ sr

♦♥♦♠s ttrs

t ♥ rstr♦♠ r♥ t① rts ♥ ♥♦♠ ♥qt② ♥ ②

♠♦ ♠r♥ ♦♥♦♠

♥r♦♥ ♥ t ♠♣r trs♠ ♣♣t♦♥s t♦ rt② ♥ rr♥ q

♥ ♦r♥ ♦ P♦t ♦♥♦♠②

♥r♦♥ ♥ P♥ rt ♥ ♥ r tt② st♥

♥ ③ s ♥♦♦ ♦ P ♦♥♦♠s ♣tr sr ♥ t

rr♦ ♥ s sr♥ t r ♠r♥ t① rt r♦♠ t

♥ ♥♦♠ t① ♦r♥ ♦ s♥ss

é♥♦ ① ♥ t♦♥ ♣♦② ♥ tr♦♥♦s♥t ♦♥♦♠② ❲t s

♦ rstrt♦♥ ♠①♠③ r♦t ♥ ♥② ♦♥♦♠tr

r♠♥ rr② ♦♥t♥ ♦r ♦s♦ ♣r♦t♦♥ ♥ t ♥t♦♥ ♦♥ts

♥ ♣t r② ♦ rr♥t s♥ss

♥ ❨ ♠ ♦♥ ♥ ♦rs♦♥ ♥tr♣rt♥ ♦r s♣♣② rrss♦♥s

♥ ♠♦ ♦ ♥ ♣rtt♠ ♦r ♠r♥ ♦♥♦♠

tt② r♠♥ ♥ ③ ❯s♥ r♥s ♥ ♥♦ r♦ss ♥♦r

♦♦s t♦ ♥♦r t ♠♣ts ♦ t t ♦♥ r♥♥s ♠r♥ ♦♥♦♠

♦ ❲ ♥ r♥s ① trt♠♥t ♦ ♦♥r ♦♣ ♦s♥ ♥ t

♥qt② ♦r♥ ♦ r♦♦♥♦♠s

♦♥ ① ♦sts ♥ ♦r s♣♣② ♦♥♦♠tr

♦♦② ♥ Prs♦tt ♦♥♦♠ r♦t ♥ s♥ss ②s ♥ ♦♦②

r♦♥trs ♦ s♥ss ② sr ♣tr Pr♥t♦♥ ❯♥rst② Prss

r ❲ ♥s ♥ P♣s ♥trt♥ ♠r♦s♠t♦♥ ♠♦s ♦

t① ♣♦② ♥t♦ ♠r♦♦♥♦♠ ♠♦ ❲♦r♥ P♣r

str② ❲ ♥ ♦ r♥ ♥♦♠ t① rts ♥ ♦♥♦♠ r♦t ♥ ♦♣♥

♦♥trs r♦♣♥ ♦♥♦♠

♦st ♥ ❩r t♦♥ ss♦rt ♠t♥ ♥ ♦s♦ ♥♦♠

♥qt② r sr ♥ ♦ ❨♦r t ♣♦rts

ss ♥ ♠♥ ♦r s♣♣② rs♣♦♥s t♦ t r♥ ♥♦♠ t① rt

rtr② ♦r♥ ♦ ♦♥♦♠s

r♥á♥③❱r ♥ rr ♦♥s♠♣t♦♥ ♥ s♥ ♦r t ② ♦

♠♣♦rt♥t r ♦♥s♠r rs r♦♦♥♦♠ ②♥♠s

♦rt♦ ♥ ❩♥ ♥t♦♠② ♦ t rt ♦r s♣♣② stt②

♦ ♦♥♦♠ ②♥♠s

r♥ r ♦♥s r ♥ P ♦t♦r t ♦r t ♥ sst

♠t♦♥ t t ♥ ♦ ♦♥♦♠ Prs♣ts

rs ♦s♥ t①t♦♥ ♥ ♣t ♠t♦♥ ♦r♥ ♦ ♦♥tr② ♦

♥♦♠s

♦ ♥ P trss t r ♥♦♠ t① ♥t♦♥s ♥ ①♣♦rt♦r②

♠♣r ♥②ss t♦♥ ① ♦r♥

r ♥ r♣s t ♦ sr♦♥♥ ♠srs ♦♥ ♦r s♣♣②

♥ ♥st♠♥t ♦♥srt♦♥s ♦r t① r♦r♠ ♦♥rss♦♥ sr r

r♥♦♦ ♥r ♦r♦ ♥ ♥t♦s rr② ②♦r ss♦rt

♠t♥ ♥ ♥♦♠ ♥qt② rs rs♦♥ ♠r♥ ♦♥♦♠ P♣rs Pr♦

♥s

r♥♦♦ ♥r ♦r♦ ♥ ♥t♦s ♥♦♦② ♥ t ♥♥

♠② ♥ ♠♦ ♦ ♠rr ♦r t♦♥ tt♥♠♥t ♥ ♠rr ♠

♦r♦r ♣rt♣t♦♥ ♠r♥ ♦♥♦♠ ♦r♥ r♦♦♥♦♠s

rr ❲ ♥ rt♥ ♦s ♦s♥ t ♠ s ss q t r♦ ♦

r ♦♦s ♥ t strt♦♥ ♦ t r sr ♦r ❲♦r♥ P♣r

♥r ②s③ ♥ ❱♥tr ①t♦♥ ♥ ♦s♦ ♦r s♣♣②

♦ ♦♥♦♠ ts

♥r ②s③ ♥ ❱♥tr ♥♦♠ t①t♦♥ ♦ s ♦s♦s ts ♥

♣r♠tr st♠ts ♦ ♦♥♦♠ ②♥♠s

t♦t t♦rstt♥ ♥ ❱♦♥t ♣t♠ t① ♣r♦rsst② ♥ ♥

②t r♠♦r rtr② ♦r♥ ♦ ♦♥♦♠s

r ♥ r♥r ♠♦rt ♥trst t♦♥ ♥ ts ♠♣t ♦♥

♦♠♦♥rs♣ s♦♥s ♦ ♦♥♦♠s ♥ ttsts

♦tr rr ♥ t♣♥ ♦ ♦s t① ♣r♦sst② ♥ ♦s♦

tr♦♥t② t r rs ❲♦r♥ P♣r

t ❱♥tr ♥ ❨r♦♥ ♦rs ♦ t♠ ♥qt② ♠r♥

♦♥♦♠

♥tr♥ r♥ sr ♦♠♣♥ st♠ts ♦r t r♥ ♥♦♠ t① rt

♠ ♦♥ rtr♥s sr ♥②ss ttsts ♣♦rt Pt♦♥

♦♥t ♦♠♠tt ♦♥ t①t♦♥ ♠♠r② ♦ ♦♥♦♠ ♠♦s ♥ st♠t♥ ♣r

ts ♦ t st ♦ t ♦♥t ♦♠♠tt ♦♥ t①t♦♥ ❳

♦♥t ♦♠♠tt ♦♥ t①t♦♥ st♠t♥ ♥s ♥ t r ♥ ♥♦♠

t① sr♣t♦♥ ♦ t ♥ t① ♠♦ ❳

st♠t t ts ♦ t ♦♥r♥ r♠♥t ♦r r tt① ts ♥

♦s t ❳

♦♥t ♦♠♠tt ♦♥ t①t♦♥ st ♦ ①♣r♥ r t① ♣r♦s♦♥s

❳

♥ ♥ ♦rs♦♥ r♦ ♥ ♠r♦ ♦r s♣♣② stts rssss♠♥t

♦ ♦♥♥t♦♥ s♦♠ ♦r♥ ♦ ♦♥♦♠ trtr

♥ P ♥ ❲s ♦r s♣♣② r♦s ♦ ♠♥ ♣t ♥ t ①t♥s

♠r♥ ♦♥♦♠ ♦r♥

♥ P♦ssr ♥ ♦ Pr♦t♦♥ r♦t ♥ s♥ss ②s

♥ ♣♣♥① ♦♠♣tt♦♥ ♦♥♦♠s

rr ♥ r ② ♠ ❯s ♠♦t♦♥ ❲♥ ♥ ❯♥♠

♣♦②♠♥t ♥ r♦♠ ♦♥t♥ t ♠r♥ ♦♥♦♠ P♣rs Pr♦

♥s

♥ P ♥ ♦③♥♦ ♣r ①♣♥♥ ❲♦r ❯♥rst♥♥ r♥s ♥

♦♥ ❲♦r ♦rs ♠♦♥ ❯ ♥ ♦r♥ ♦ ♦r ♦♥♦♠s

②♥ s♥ss ②s ♥ rt ♦r ♠rt tt♦♥s ♥ ♦♦②

r♦♥trs ♦ s♥ss ② sr ♣tr Pr♥t♦♥ ❯♥rst② Prss

ttr ♦♠♣rs♦♥ ♦ st♠ts ♦ ♣rs♦♥ ♥♦♠ ♥ rs st♠ts ♦

st r♦ss ♥♦♠ r ♦ ♦♥♦♠ ♥②ss r② ♦ rr♥t s♥ss

❲ ❨♥ ♥ ❨♦ ♦s♥ ♦r t♠ ♥ ♦r t ②

strtr st♠t♦♥ ♥tr♥t♦♥ ♦♥♦♠

❲ ♥ P rt Pr♦rss t①t♦♥ ♥ ♦♥r♥ r♦t ♠r♥ ♦♥♦♠

st ♠rt ♦r rt ♥ ♦r♥ ♦ ♦♥♦♠ Prs♣ts

♦s ♥ ♦tst♥ r♥ ♥♦♠ t① rt ♥ ♦tt

♦♥♦♠s ♦ ♥sst r♥sr Pr♦r♠s ♥ t ❯♥t tts ♣tr ❯♥rst②

♦ ♦ ♦♦s

P♥ r♦ ♦ t tr♥s♦r♠t♦♥s ♥ ♣♣t♦♥ t♦ st♠t♥ t t

♦ t① ♥♥ts ♦♥ s♥ ♦♥trt♦♥s t♦ ♦♥♦♠ ♥②ss ♥ P♦②

Pr r♥ ♥♦♠ t① rt ♣rt♣t♦♥ rt ♦r t① ②r ♥tr♥

♥ r sr t♥

♦rs♦♥ ♥ ❲♥s r♦ ♥ ♠r♦ stts ♥ ② ♠♦ t

t①s ♦r♥ ♦ ♦♥♦♠ ♦r②

♦rs♦♥ ♥ ❲♥s ♥ ♦♥♦♥①ts tr♠♥t ♥ t stt② ♦

♦r ♣♣② ♠r♥ ♦♥♦♠

♣rt P ♥ ❩♥ st♥ r♥♥s ♥ ♦rs ♣r♦s ♦r♥ ♦

♦♥tr② ♦♥♦♠s

❱♥tr t t① r♦r♠ q♥ttt ①♣♦rt♦♥ ♦r♥ ♦ ♦♥♦♠ ②♥♠s

♦♥tr♦

❲♥ qsts ♥ t ❯ Pttr♥s ♦ts ♥ ① P♦② P tss ①s

❯♥rst②

♣♣♥s ♦r ♥♥ Pt♦♥

rt♦♥

♦♥① P♦② Pr♠tr ❱s ♥ rts

♠♦r♣s

srt t♠ ♥ t ♠♦ ♣sss t ♥ ♥♥ rq♥② t t ♥ ♦ ♣r♦ ♥s j = R = 40 rtr ♥ ♥ t♦ ♠♦rtt② rs t♦s j = J = 66 t rt♥t② ♦ j = 1 s st t♦ ♦rrs♣♦♥ t t s♦tt ♠♦ j = 40 ♦rrs♣♦♥s t t ♥ ♠♦ j = 66 ♦rrs♣♦♥st t

♣♦♣t♦♥ r♦t rt s st t♦ υP = 0.0075 s t r ♥♥ ❯♣♦♣t♦♥ r♦t rt ♣r♦t ② t ♥ss r ♦r ②rs rt♦ t♥♦♦ ♣r♦rss s st s♦ tt t st② stt r♦t rt ♦ rt ♦t♣ts q♥t t♦ t r ♥♥ r P r♦t rt ♦ 1.93% ♣r♦t ② t♦♥rss♦♥ t ♦r ②rs ♥ rt ♦t♣t r♦st rt υP + υA ♥ ♠r♦♦♥♦♠ st② stt st υA = 0.0118

♦♥t♦♥ sr ♣r♦ts πj r st t♦ πj = 1 ♦r s j < R s ♦s♦s♥ t ♠♦ ♦♥② ♥ t♦ ♠♦rtt② rs ♣♦♥ rtr♠♥t ❲ rt t ttr♦♥t♦♥ sr ♣r♦ts s♥ t ♦ rt② ♠♥strt♦♥s t

r ❲ ♦♠♣t t r ♦r ♠s ♥ ♠s ♦r ♥ t♦ ♦t♥ s♥ r♥t srs ♣♣② t♦ ♦s♦s

♠sr ♦ ♦s♦s Ωz,fj s ♦♥strt ♥ ♦r st♣s rst ♥ t ♦♥st♥t

rt ♦ ♣♦♣t♦♥ r♦t ♥ t♠♥r♥t ♠♦rtt② ♦♥strt stt♦♥r② ♣r♦ ♦ ♦s♦s s Ωj+1 = (Ωjπj)/Υ

P ①t t ♦♥t ♥st② Ωfj s ♦♥strt

rt② ② ♦♠♣t♥ t srs ♦ ♦♥t ♥ ♥♦♥♦♥t t① ♥ts ♦t ♦ t♦t ♥ts ♦r tr♦ ♣r♦t ♦r s♥ t r t ♥st② Ωz s♦♥strt ♥♣♥♥t② ♦ ♥ ♠② ♦♠♣♦st♦♥ ♥r t ss♠♣t♦♥ tt t♠ss ♦ ♠② ♦♠♣♦st♦♥ r♦♣ s q♥t ♦r ♦ t nz ♣r♦tt②t②♣s ♠♣s Ωz = 1/nz ∀z ∈ Z ♥② Ωz,f

j = ΩfjΩ

z s ♥♦r♠③ t♦ t♣r♦♣rt②

∫

J

∫

Z

∑

f=s,m

Ωz,fj = 1

♦♥strt ♥ ts ♠♥♥r t ♠sr ♦ ♦r ♣r♦tt② t②♣ ♦r t ♣♦♣t♦♥ ♣r♦ s ♦♥st♥t t sr ♦ s♥♠rr ♦s♦s s r♥t

r♠ Pr♦t♦♥ ♥♦♦② ♥ ♦s♥

♦ t t sr ♦ ♦t♣t ♥ ♣r♦t♦♥ ♦r ♣rt ♥ ♣ ♣t rs♣t② ♦rr♦ r♦♠ t ♠t♦ ♦ ♦♦② ♥ Prs♦tt ♦ts t ♠

❯♥ss ♦trs ♥t ♣r♦t♦♥s r♦♠ t ♦♥rss♦♥ t r r♦♠ t

♥ ♦♥♦♠ t♦♦ t♦ ♣r t♠ ssr♣t s ♦♠tt ♦r ♥r ①♣♦st♦♥s s ♥tr ss♠♣t♦♥ s s sr ♥ t♦♥ ♣r♦tt② r♦♣s

t♥ t♦ r♣rs♥t t♠ ♦r ♥♦♠ q♥ts

♦s ♦♠♣♦♥♥ts ♦ rt ♥♦♠ ♥ ♣r♦♣♦rt♦♥ t♦ t♦rs sr ♥ ♠sr♦t♣t ❯s♥ t r♦♠ t t♦♥ ♥♦♠ ♥ Pr♦t ♦♥ts P ♦ tr ♦ ♦♥♦♠ ♥②ss ♦r rst t 36.17% s t ♦♥t sr♦ P ♦r ♦t ♣ ♥ ♣rt ♥♦♥rs♥t ♣t ❲ t♥ r♣t t t♦♥ ♣rt ♥♦♥rs♥t ♣t ♦♥② ②s ♣rt ♣ts sr ♦ ♣r♦t♦♥α = 0.3265 ♥② t ♣ ♣ts sr ♦ ♣r♦t♦♥ t♦ t rs ♦ t♦♥t sr ♥ ♣rt ♣ts sr s♦ tt g = 0.0352

s ♥ r♥á♥③❱r ♥ rr rts ♦ ♦♥♦♠ ♣rt♦♥ ♦♥♣r♦t ♣t ♥ ♦s♥ ♣t r t s♥ t st② stt ①♣rss♦♥ ♦rt ♥st♠♥t t♦ ♣t rt♦ ♦r ♣r♦t ♣rt ♣t ts ①♣rss♦♥ s IK/K =(ΥAΥP − 1 + δK) ♦♥ ♦r t ♣rt♦♥ rt ②s δK =

(

(IK/K)−ΥAΥP + 1)

❯s♥ t r ♥♥ ♥st♠♥t ♦s ♥ st♦s ♦ ♣rt ♥♦♥rs♥t ♣ts r♣♦rt ② P ♦r ②rs ②s δK = 0.0799 ❯s♥ t ♥♦♦s①♣rss♦♥ ♦r ♣ ♣t t r t♦ ♥ r stt ♥ ♦♣t ♦♠♣t δG = 0.0317 ♦r t s♠ ♣r♦ ♦r ♦s♥ ♣t tt ♣rt♦♥ rts s♥ t r ♥♥ ♥st♠♥t ♦s ♥ st♦s ♦ ♣rtrs♥t ♣t ♥ ♦♥s♠r rs s r♣♦rt ② P ♦r ②rs ①♣rss♦♥ ♦r t ♦♥r♦♣ ♦s♥ ♣t ♣rt♦♥ s t s♠ s tt ♦r♣rt ♣t ♦t♥s δo = 0.0555 ♣rt♦♥ rt ♦♥ r♥t ♦s♥♣t rs r♦♠ ♦♥r♦♣ ♦s♥ ♣t s r♥t ♦s♥ ♥st♠♥ts ss♠ t♦ s ♦♥t♠♣♦r♥♦s② t ♥st♠♥t ♦s ♦r r♥t ♦s♥♣t t ♥st♠♥t t♦ ♣t rt♦ s Ir/Hr = (ΥAΥP − 1 + δr)/(ΥAΥP ) st ss♦t ♣rt♦♥ rt δr = ΥAΥP ((Ir/Hr)− 1) = 0.0570

♥♥ ♥tr♠r② s qrt ♦sts ♦ ♣t st♠♥t q t♦ Ξt =ξK

2(Kt+1

Kt− ΥPΥA)

2Kt ♥ t rts ♦ ♣♦♣t♦♥ r♦t ♥ t♥♦♦ ♣r♦rss

ts st♠♥t ♦st ♥t♦♥ s ♣r♠tr③ ② ξK ♦r ♣r♣♦ss ♦ t s♠t♦♥s s st t♦

♦♦♥ rs r♥á♥③❱r ♥ rr ♥ ♦ ♥ r♥s st t ♠♥♠♠ ♦♥♣②♠♥t rqr ♦♥ ♦♥r♦♣ ♦♠ ♣rss t♦ γ = 0.20 s r s♦ ♦s② ♦rrs♣♦♥s t♦ t ♠♥ ♦♥t♦ rt♦♦ ♦r ♦♥r♦♣ ♦s♥ ♥ts ♠♥tr t♥ s r♣♦rt♥ t ♥ss rs ♠r♥ ♦s♥ r② rtr♠♦r s ♥ rr ♥rt♥ ss♠ tr♥st♦♥ ♦sts ss♦t t ♥♥ ②♦r ♦s♥ stts♥t r s②♠♠tr ♦r ②♥ ♥ s♥ ❲ t t♦rs ♥ ♠♥ ♦sts ♦ 7% ♥2.5% ♦ ♦s♥ ♦r s♥ ♥ ♣rs♥ rs♣t② ♦♥srt② ♦♦st ♠♣♦♥t s♦ tt φo = φr = 0.05

♦ rstrt ♦s♦s r♦♠ ♣rs♥ ♥s② s♠ rs♥s ss♠ trs ♦r ♦♥ ♦♥ t s♣♣♦rt ♦ r♥t ♦s♥ r ♥ ♦♥r♦♣ ♦s♥ o stt 0 < r ≤ o ♦r ♦♥ ♦♥ ♦♥r♦♣ ♦s♥ s st t♦ trt ♦♠♦♥rs♣ rt♦ ♦ s r♣♦rt ♦r ② t ♠r♥ ♦s♥ r②t♥ t rt♦ ♦ ♦s♥ t♦ ♦♦ s♣♥♥ ♦r t ♦st ♦ ♦s♦s ♥ s r♣♦rt ② t r ♦ ♦r ttsts ♥ t ♦♥s♠r ①♣♥tr

❲ ♦♦② ♥ Prs♦tt t t♦r ♥♦♠ sr ♦ P ♦♦ tr ♠t♦♦♦②t♦ ♥st t t♦r ♥♦♠ srs ♥ P

❲ ♦♥sr ♠♦s ♦♠♣♦♥♥ts t♦ ♣r♦♣rt♦rs ♥♦♠ t sttst sr♣♥② t①s ♦♥♣r♦t♦♥ ♥ ♠♣♦rts ♥ t rr♥t sr♣s ♦ ♦r♥♠♥t ♥tr♣rss ss sss ❲ t ttrtr r ♦t♥ ① ♥ t♥ t♦r ♥♦♠ srs ♥ t♠ s♦ tt t rt♥♦♠♦t♣t q♥ ♥ t ♠♦ ♠♣s ♦ ♦t♣t ♦♥sst♥t t ♠sr P

r② t ♦r ♦♥ ♦♥ r♥t ♦s♥ s st t♦ t♠s t r♠♥♠♠ ♦♥s♠♣t♦♥ s r♦ss ♠② ♦♠♣♦st♦♥

♦s♦ rtrsts ♥ Prr♥s

rt t ♦s♦s s♦♥t tr tt② s st t♦ trt t ♦sr ♣rts♥ss ♥st♠♥t t♦ P rt♦ ♦ s t r♦♠ P ♦r ❲ st tst s♦♥t t♦r s β = 0.94 t♦ trt ts r ♥r ♦t t① s②st♠s

♥ ♦r ♣r♦tt② ③z,fj s ss♠ t♦ ♦♥sst ♦ t♦ ♥♣♥♥t ♦♠♣♦♥♥ts (i) ♥ r②♥ ♦♠♣♦♥♥t ③j ♥ (ii) ♥ ♥r♥t ♦♠♣♦♥♥t tt ♣♥s

♦t ♦♥ ♣r♦tt② t②♣ ♥ ♠② ♦♠♣♦st♦♥ ③z,f s♦ tt ③z,fj ≡ ③j③z,f ❲ t t

♥♠r ♦ ♣r♦tt② t②♣s nz = 5 s♦ tt t②♣ z ♣♣r♦①♠ts r ♥♥♦s♦ ♦r ♥♦♠ q♥ts ♥ ♦s♦ ♦ ♣r♦tt② t②♣ z r♠♥s ttt②♣ ♦r tr t♠ ♦r rt♦♥ s ♦♥sst♥t t t ♥♦t♦♥ ♦ ❵t♠ ♦rr♥♥s q♥ts s s t♦ ♥tr s♦rt♥ ♦ ♦s♦s ② tr ♦r ♥♦♠rtrsts r ♥ ♦r ♥♦♠ t♦ t s♠ ♦ P♦♠♣r ♥♦♠ ♦♥♣t ♥ (1− α− g) sr ♦ ♣sstr♦ s♥ss ♥♦♠,

♦ rt t r②♥ ♦♠♣♦♥♥t ♦ ♦r ♣r♦tt② ♦♣t t s♠♦♦t♥♠r ♣r♦s st♠t ♥ ♣rt ♥ ❩♥ ♦r ♥s ♥♦r♠③♥ t ♠♥ t♦ ♥t② ♦r t ♥r♥t ♦♠♣♦♥♥t ♦ ♦r ♣r♦tt② st③z,f ♦r z = 1, . . . , 5 ♥ f = s,m ♥♦♥♦s② s♦ tt r ♥♥ ♦r ♥♦♠♦r s j = 1, . . . , R ♥ t ♠♦ ♠ts t s ①tr♣♦t s ♦r ♥♥ ♦r ♥♦♠ ♦r s ♦r rs♣t r♦♣ ♥ ♥ ♣rs♥t s♥ qr♠ ❲ ♦t ♣♦t♥t ♦rrs ♥ ♠rr ♦s♦s ts♠ ♥ ♦r ♣r♦tt② tr♠ ③z,fj tr s ♥ ①♦♥♦s ♣r♦tt② µz t♥ ♣r♠r② ♥ s♦♥r② ♦rrs s s t♥ t♦ r♣rs♥t t ♦r②r♥♥s ♦ s♦♥r② ♦rrs rt t♦ ♣r♠r② ♦rrs ♥ s ♦♠♣t s♥ tt♦t ♥ s♥ss ♥♦♠ q♥ts ♦ ♣r♠r② rs ♥ tr s♣♦ss s r♣♦rt② t ①♣♥trs P♥ r② P ♦r

♥ ♦r s♣♣② n s ♥s ♥ s ss♠ t♦ ♦rrs♣♦♥ t tr t♠ ♠♣♦②♠♥t nFT ♣rtt♠ ♠♣♦②♠♥t nPT ♦r ♥♠♣♦②♠♥t ♦♦♥ t ♦♥sr t♠ ♠♣♦②♠♥t t♦ ♦rrs♣♦♥ t t st ② ♦rs ♦ ♦r ♥♣rtt♠ ♠♣♦②♠♥t t♦ ♦rrs♣♦♥ t t st t ♥♦ ♠♦r t♥ ② ♦rs♦ ♦r ❯s♥ t r♦♠ t ♥ tt ♥ t ♠♥ t♠ ♥ ♣rtt♠ ♠♣♦②s ♦r ♥ ♦rs ♣r rs♣t② r q♥t t♦♦t ♥ ♦r♥ ♦rs ♣r ②r s t r♣♦rts ♥ t ♠r♥

♠ ❯s r② tt t r ♥ s♣♥s ♦t ♦rs s♣♥ ♣r②r t♠ ♥ ♣rtt♠ ♦r ♦rrs♣♦♥ t 41.6% ♥ 21.0% ♦ ♥ ♦rss♣♥t ♦r♥ ❲ tr♦r ♥♦r♠③ ♥ t♦t t♠ ♥♦♠♥ts t♦ ♥t② ♥

♦s ♥♦t r♣♦rt strt♦♥ rtrsts ♦ P ♥♦♠ ♦ ♣♣r♦①♠tt sr P ♥♦♠ q♥ts r ❵P♦♠♣r ♠sr s♥ t ②♥ t♦ ♥♦♠ (i) ♦♠t ♣② (ii) ♠♣♦②rs sr ♦ t t① (iii) rr ♦♠♣♥st♦♥ (iv) ♠♣♦②rs sr ♦ ♦♠♣♥st♦♥ (v) ♠♣♦②r ♣r♦ ♣♥♥t r (vi)♠♣♦②r t♥sr♥ ♦♠♣♥st♦♥ (vii) ♠♣♦②r ♦♠♣♥st♦♥ ♥ (viii) ♠♣♦②r ♥sr♥ ♦♠♣♥st♦♥ ttr ♦r ♦♠♣rs♦♥ ♦ ♥♦♠ ♥ P ♥♦♠

♣sstr♦ s♥ss ♥♦♠ ♦♥sr r ♥s ♥♦♠ ♦s r♦♠ s♦ ♣r♦♣rt♦rs♣♣rt♥rs♣s ♦r♣♦rt♦♥s stts ♥ trsts ♥ r♥ts ♥ r♦②ts

♦ ♠t t ♠♦r♣s ①♣t② ♠♦ rstrt ♦r tt♥t♦♥ t♦ ♦s♦s ♦ ts s♦ r ♥♦t r♥ rtr♠♥t ♥♦♠ ♣♥s♦♥ ♦r s♦ srt② ♥♦♠

st nFT = 0.416 ♥ nPT = 0.210 ♦r stt② ♦♥ts ψs, ψm,1, ψm,2 ♥ ① ♦st ♦ ♠♣♦②♠♥t ♣r♠

trs φs, φm r st s♦ tt ♠♦♥rt ② ♦r s♣♣② ♣♣r♦①♠ts tstrt♦♥ ♦ ♠♣♦②♠♥t stts ② r♥r t②♣ s♥ ♠rr ♣r♠r② ♦r ♠rrs♦♥r② s ♦sr ♥ t P ♦r ♥ r♣♦rt ♥ ①♦st ♦ ♠♣♦②♠♥t ♣r♠tr r② tr♠♥s t ♦r♦r ♣rt♣t♦♥ rt ①t♥s ♠r♥ stt② tr♠♥s t s♥stt② ♦ ♦rrs ♠♦♥ ♥t♦ ♦r♦t ♦ ♠♣♦②♠♥t ♥ rs♣♦♥s t♦ ♥s ♥ t trt① rt s tr♠♥ ♥♦♥♦s② ② t strt♦♥ ♦ rsrt♦♥ s ts ♣rt② ♣♥s ♦♥ t① ♦sts ss♦t t ♦r♥

♥t♥s ♠r♥ stt② tr♠♥s t s♥stt② ♦ ♦rrs ♦♦s♥t♦ s♣♣② ♠♦r ♦r ss ♦rs ♦ ♦r ♥ rs♣♦♥s t♦ ♥s ♥ t trt① rt♦♥t♦♥ ♦♥ r② ♥ ♠♣♦② ♣♥s ♦♥ t tt② ♥t♦♥ rtr ♣r♠trs ζs, ζm,1, ζm,2 ♦♥② ♥ ♠♦s ♦ ♦♥t♥♦s ♦r ♦ ♥ ♠♦s ♦ srt♦r ♦ s s t ♦♥ s r ♥ t s♦ tt ts ♣r♠trsr r② ♥rt t♦ t ♥t♥s ♠r♥ stt② ♦rs♦♥ ♥ ❲♥s ♥ ♥ ♥ ♦rs♦♥ s♦ tt r s ♦ ts ♣r♠trs ♠♣② ttrt ♠♣♦②♠♥t tt♦♥s ♣♥ ♠♦r ② ♦♥ ♥s ♥ t rt♦♥ ♦♦r♥ rtr t♥ ♥s ♥ ♦rs ♦r ♠♣♦② st♠t stts♦r s♦♥r② r♥rs r♦♠ tr ♠♦ s♠t♦♥ r s②st♠t② r s♣t ♥ t s♠ rtr ♣r♠tr s t ♣r♠r② r♥r Pr♣s ♠♦st ♠♣♦rt♥t② ♦rrt♦♥ ♣r♣♦ss ♠♦♠♥t ♦♥ t ♥t♥s ♠r♥ s♦ ♦♥t ♦r ♦♥tr♦ rt ♠♣♦②♠♥t tt♦♥s ♦♥ r ♦r s ♦rt♦ ♥ ❩♥ ♥ ts ♥t♦ ♦♥srt♦♥ ♦♦s t s♠ rt② ♥♦r♠ ♦r ♦rrs ♥ st ζs = ζm,1 = ζm,2 = 5

♠♦♥t ♦ ♦rs s♣♥t ♦♥ ♦♠ ♣r♦t♦♥ ① ♥rs rt♦♥s♣ t♦t ♠♦♥t ♦ ♦r ♦rs ❲ s t ♠r♥ ♠ ❯s r② t♦ ♥ tr ♦s♦r ♦rs ♦r t♠ ♣rt t♠ ♥ ♥♠♣♦② ♥s t♠♦rrs r♣♦rt s♣♥♥ ♦rs ♣r ② ♣rt t♠ ♦rrs r♣♦rt ♦rs ♣r② ♥ ♥♠♣♦② ♥s r♣♦rt ♦rs ♦r♠③♥ ♥♦♥s♣t♠ t♦ ♥t② ②s t ♦♦♥ ♠♣♣♥ ♦r ♦♠ ♦r t♠ s ♥t♦♥ ♦ ♦r♦rs ♦r s♥s ♥ ♠rr ♥

N = [0.000, 0.210, 0.416] → Nh = [0.114, 0.100, 0.041]

♠♣r② t ♦ ♦♠ ♣r♦t♦♥ s ♥ ♠sr ② ♠t♣②♥ ♦rs

P r♣♦rts t ♠♣♦②♠♥t stts ♦ ♥ ♥ ♠rr ♦s♦ t ♦s ♥♦t s♣②♦ s t ♣r♠r② r♥r tr t♥ rr♦♥♦s② s♥ ♥r s ♥ ♥t♦r ♦ ♣r♠r② ♦r s♦♥r②r♥♥s stts ♦♥sr t ♠♦♥t ♦ ♦rs ♦r ♦t ♥s r ♥♠♣♦② ♦♥srt ♣r♠r② r♥r t♦ t ♦♥ ♦ s ♥♠♣♦② ♦t ♥s r ♦r♥ ♣rt t♠ ♦r ♦♥s ♦r♥ ♣rt t♠ ♥ ♦♥ s ♥♠♣♦② ♦♥sr t ♣r♠r② r♥r t♦ t ♦♥ ♠♣♦②♣rt t♠ st② t st ♦♥ r♥r s ♠♣♦② t♠ ♦♥sr t ♣r♠r② r♥r t♦ t♦♥ ♠♣♦② t♠ ♦♥sst♥t ♥t♦♥s ♦ t♠ ♣rtt♠ ♥ ♥♠♣♦② r s ♥♦♥strt♦♥ ♦ ts trts

♥r ♦rt r ♦♦ ♥ tt ♦ ♠r♥ ♠ ❯s r② t ①trtr ❱rs♦♥ ♦ Pr ❯♥rst② ♦ r②♥ ♥ ♥♥♣♦s ❯♥rst② ♦♥♥s♦t tt♣s♦♦r❱

❲ s♠♠ ♦r rs ♦r ♦♠ ♦r rs ♦s♦r ♦♦ ♣r♣ ♥ ♥♣ ♥tr♦r♠♥t♥♥ ♥ ①tr♦r ♠♥t♥♥ ❲ ♦s rs tt ♥ rs♦♥② ♦ts♦r ♦r ♣②♥ s♦♥ ♠♦st ♦rs rtr t♥ ♦s ♦r sr tts

s♣♥t ♦♥ ♦s♦r ② t rt ♦ ♦♠st ♦rrs s r♠♥ ❲ t s♠r ♣♣r♦ t♦ t♥ t ♦ ♦♠ ♣r♦t♦♥ s ♥t♦♥ ♦ ♥♦♥♦r♦rs

ch(nhj ) =

wt③1,snh

j f = s

wt③1,s(nh,1

j + nh,2j ) f = m

r wt③1,s s t r rt ♦r t ♦st ♣r♦tt② t②♣ s♥ ♦s♦

r ①♣♥ss t t ♦r♠ κz,fj ≡ ccz,fνz,fj nj r ♦r s♣♣② s tt t q♥tt② s♣♣ ② t s♦♥r② ♦rr ♦r ♠rr ♦s♦s ♥ tr ♥♠r ♦ ♣♥♥ts ♥r t♥ ♦s♦ νz,fj st t r ♦st

s ♣r♠tr ccz,f ①♦♥♦s② s♦ tt r ①♣♥ss ♦♥ r ♦r (z, f)♠♦r♣ ♥ ♦r s♣♣② s t s t ♠♣♦②♠♥t trts sss ♦♠t t♦s s ♠♣t ② t ♦r

♥ t s♣t♦♥ ♦ t ♥♦♥♦s♥ ♦♥s♠♣t♦♥ ♦♠♣♦st ♥ qt♦♥ t rt ♦♣t♠ q♥tts ♦ ♦r♥r② ♦♥s♠♣t♦♥ ♥ rt ♥ ♥ ①♣rss ♦r t♦s ♦s♦s ♥♦t t♠③♥ t① t♦♥s s

cgjcij

=

(

1− θz,f

θz,f

)

t (cg/i)z,f ♥♦t r rt ♥ s ♣r♦♣♦rt♦♥ ♦ ♦r ♥♦♠ trts ♦r (z, f) ♦♠♥t♦♥ s ♦♠♣t r♦♠ t ♦r ♥ t ♦♥s♠♣t♦♥♦♠♣♦st ♥t♦♥ ♥ ♣r♠tr③ ♥♦♥♦s② ② stt♥ θz,f s tt

θz,f =

(

1 + (cg/i)z,f∑R

j=1 iz,fj

∑Rj=1 c

i;z,fj

)

−1

♠♣s tt ♥ s♥ qr♠ t ♠♦ r♣r♦s rt ♥ trts♦♥ r ♦r t ♦r♥ ♣♦♣t♦♥ ❲ ts strtr ♦s ♦r t ♠♦ t♦♥rt t ❯s♣ ♣ttr♥ ♦ r ♦s♦ ♥ ♦r ♥♦♠ st tr♥ ♥ ♥ ♠♦♥ ♦s♦s ♦ ♥ ♠♦r♣ ♥rstt

stt② ♦ ssttt♦♥ t♥ ♦s♥ ♥ ♥♦♥♦s♥ ♦♥s♠♣t♦♥ s stt♦ η = 0.487 s st♠t ② t ♥♦♥♦s♥ ♦♥s♠♣t♦♥ ♣rr♥♣r♠tr σ s st t♦ trt t rt♦ ♦ ♣rt s♥ss ♥st♠♥t t♦ t♦t ♣rt ♥st♠♥t ♦ s t r♦♠ t P ♦r ♦♥ ♦♥ ♣r♠ss ♦r♥r②♦♥s♠♣t♦♥ s i s st t♦ ♥ rtrr② 5% rr t♥ t ♠①♠♠ ♠♦♥t ♦♦♠ ♣r♦t♦♥ ♦♥s♠♣t♦♥ tt ♠② ♦t♥ r♦♠ ♦♦s♥ ♥♠♣♦②♠♥t s♦tt ts ♦♥str♥t ♥ ♣♦t♥t② ♥

♦r♥♠♥t P ♣t ♥ t

Pr♦t ♣ ♣t ② t r ♦r♥♠♥t ♥ t stt ♥ ♦ ♦r♥♠♥t r ♦t st ♥♦♥♦s② s♦ tt ♥ ♣rs♥t s♥ t② ①t ♦18.4% ♥ 55.1% rs♣t② ♦ rt ♦t♣t s rts ♥ r ♦r r♦♠ P ♥ ts trt ♥ t ss♠♣t♦♥ tt ♣ ♣t r♠♥s① ♥r ♣r♦♣♦s s♠t♦♥s t ♦ ♥st♠♥t ♥ ♣ ♣t ♥ ♦♠♣t r♦♠ t qt♦♥ ♦ ♠♦t♦♥

♣r♦ts tt r t ② t ♣ ss ♥♥ ssts rt t♦

P r♦ r♦♠ 65.7% ♥ s ②r t♦ 84.0% ♥ s ②r s ♦ tsr ♣r♦t ♥rs rt r t ② t ♣ ♥ t ♣rs♥ts♥ s♦ tt s t ♣♣r♦①♠t② 74.6% ♦ rt ♦t♣t t r ♣r♦t ♦r s ②rs rtr♠♦r t ❯♥t tts rsr② r♣♦rts ♥t ♠r rsr② t♥ tt 21.7% ♦ t ② t ♣ s ②r sr ♥s t t ♥♥♥ ♦ s ②r ♥ ts ♣♦rt♦♥ ♦ ♣t ♦s ♥♦t ♥ssr② r♦ ♦t ♣rt ♣t ♥♦♥♦s② st t ♥t st♦♦ ♦r♥♠♥t t B ♥ t ♣rs♥t s♥ s♦ tt ts rt t♦ rt♦t♣t s ♣♣r♦①♠t② 58.4%

t s s♦ ♣r♦t ② t tt ♥t ♥trst ♣②♠♥ts ♦♥ r t rtt♦ P r♦♥ ♦r s ②rs ❲ tr♦r ♥♦♥♦s② st t t♥ rt ♦ rtr♥ t♦ ♣rt ♣t ♥ ♦r♥♠♥t t s♦ tt ♥t♥trst ♣②♠♥ts rt t♦ ♦t♣t ♠t t r ♣r♦t ♦ ♦rts t♠ ♣r♦

♥♦♠♥t tr♦♥t②

s s♦♥ ♥ t t ♦r ♠♦s ♦ ts ss r♥s ♥ ♥t t♦♥trt t♦ sst♥t ♣♦rt♦♥ r♥s ♥ t♠ t ♦ ♦♥t ♦r tsrt♦♥s♣ ♥tr♦ rt♦♥ ♥ ♥♦♠♥ts ♦ ♥t ♥♥ t a1 ♦r ♠♦r♣ s ♥① ② e = 1, . . . , ne ∈ E

♦ s♣② ♥♥♥♦♣r♦ ♥♦♠♥ts ♦r ♦s♦s ♥tr♥ t ♦♥♦♠② t j = 1 r rt♦s ♦ ♥♥ t t♦ ♦r ♥♦♠ Af r♦♠ t ♦♦♥strt♦♥

sinh−1(

Af)

∼ N(

µf , σ2;f)

r µf ♥ σ2;f r t♥ t♦ t s♠♣ ♠♥ ♥ r♥ ♦ t ♥rs ②♣r♦ s♥ ♦ ♥♥ t t♦ ♦r ♥♦♠ rt♦s ♦r s♥ ♥ ♠rr ♠s ② ②r ♦ ♥ t r② ♦ ♦♥s♠r ♥♥s ♦r rs♣t②xf = −0.0439, 0.0045 ♥ s2;f = 0.7464, 0.6153 tt♥ iz,f ♥♦t t rt♠ ♦r ♥♦♠ trts ♥ zj t r②♥ ♦♠♣♦♥♥t ♦ ♥ ♦r♣r♦tt② t t ♣r♦t iz,fz1 s ♥ ♣♣r♦①♠t♦♥ ♦ r ♦r ♥♦♠ ♦rt ②♦♥st ♦♦rt ♦ ♦s♦s rt♥ s♥ ♥ ♠rr ♦s♦s s s♣rtstrt♦♥s t ne = 40 r♥♦♠ rs ♦ Af ♦r ♣r♦tt② t②♣ ♦t♥♥t ♥♦♠♥ts ♦ ♥♥ t

ae,z,f1 = Afz1

(

iz,f

inz,f

)

r t ♥♦♠♥t♦r ♦♥ t rt♥ s ♥♦r♠③s ♥t ♥♦♠♥ts ② t r ♦r ♥♦♠ ♦ t st ♣r♦tt② t②♣ rtr♠♦r t t ♠♥♠♠

❲ ♦♦s t ♥rs ②♣r♦ s♥ tr♥s♦r♠t♦♥ t♦ ♣♣r♦①♠t ♥tr ♦ tr♥s♦r♠t♦♥ ♦♥ ♦r ♥t s P♥ ♦r sss♦♥

❲ ♥ ♥♥ t s ♥♥ ssts ♥s ♦ ♥ ♦♥ts s♥s ♦♥ts ♠♦♥②♠rt ♠t ♦♥ts ♦♥ts t r♦rs ♣r♣ rs rtts ♦ ♣♦sts t♦t rt② ♠t ♥s st♦s s♥s ♥ ♦tr ♦♥s s trt ♦♥ts tr ♣♥s♦♥s s ♦ ♦ ♥sr♥ trsts ♥♥ts ♠♥ ♥st♠♥t ♦♥ts t qt② ♥trst ♥ ♠s♥♦s ♦tr ♥♥ ssts ss t rt r ♥s t♦♥s ♦♥s ♥st♠♥t ♦♥s ♦♥s♥st ♣♥s♦♥s ♥♦r ♥sr♥ ♠r♥ ♦♥s ♥ ♦tr ♠s♥♦s ♦♥s

r♥ ♦ ♥♦♠♥ts ♦r ♦r ♣r♦tt② t②♣ z ♥ ♠② ♦♠♣♦st♦♥ fs t ♦r♦♥ ♦ t s♣♣♦rt ♦r t rs♣t ♠♦r♣

②z,f = argmin(ae,z,f1 )

❲ ts rt♦♥ ♥ ♥♦♠♥t ♦s ♥♦t ♥ t ②♥♠ ♦♣t♠③t♦♥♣r♦♠ ♥♦♠♥t tr♦♥t② ♦s ♥ t♦♥ ②r ♦ rt♦♥ s tt♦r ♥② r ①

①z,ft,j =

∫

E

①z,f,et,j Ωe de

r Ωe = 1ne

s t ♠sr ♦ ♥♦♠♥t e r♦r ♥ ②r ♦ ts♠t♦♥ nf×nz×ne ♦s♦s ♥tr t ♠♦ s ♦ rt♦♥ s ♠♣t♥ ♣♣♥①

②♥♠ Pr♦r♠♠♥ Pr♦♠

tt♦♥rt②

♦♥ st②stt ♥ r♦t ♣t rt ♥ ♥ rs r♦♥ t ♦♥st♥t rt ♥ ♦rr t♦ ♣♣② ♥♠r s♦t♦♥ t♥qs ♦r stt♦♥r②♦♥♦♠s ①♣rss ts rs ♥ tr♥stt♦♥r② ♦r♠ t♦ st ♦r t s♦r♦ r♦t ♦ tr ♣♦♣t♦♥ r♦t t♥♦♦ ♣r♦rss ♦r ♦t

tt♥ t t ♥t ♥♦t r tr♥s♦r♠ t♦ stt♦♥r② ♦r♠ stsst r♦tst rs

♦♥sr rt ♦r s♣♣② N ♥ st② stt r♦♥ t♦♣♦♣t♦♥ r♦t ❲ rt ♦r s♣♣② tr♦r r♦♥ t rt♦ υP t stt♦♥r② r N ♦♥st♥t r② ♦tr rt r r♦♥ ♥ st② stt t♦ ♦t ♣♦♣t♦♥ r♦t ♥ t♥♦♦ ♣r♦rssr♦r t② ♥ ♠ stt♦♥r② ② ♥ t t ♣r♦t AP ❲ ♦ ts ♥♦♥stt♦♥r② rts r♦♥ t t rt ♦ υP +υA ♥ st②stt t stt♦♥r② r ♦♥st♥t

♠sr ♦ ♦s♦s ♥ ♣r♦tt② Ω r♦♥ t rt ♦ υP s♦② t♦ ♣♦♣t♦♥ r♦t ♥ tr♦r ♥ ♠ stt♦♥r② ② ♥ ② tt♦t ♣♦♣t♦♥ r② ♦tr ♥ r r♦s t rt ♦ υA ♥ st② stt s♦② t♦ t♥♦♦ ♣r♦rss s rs ♥ tr♦r ♠ stt♦♥r② ②♥ t t ♦ t♥♦♦②

tt♦♥r② rs ♦r♠t♦♥

❲ ①♣rss t ♠♦ sr ♥ t♦♥ s tr♥ stt♦♥r② ②♥♠ ♣r♦r♠ s s♦ ♥♠r② ♥ ♦♥ s♦ rst ♣r♦r♠ ♥ ♦ rs t♦ ♠tt t rs♦♠♥s♦♥t② ♣r♦♠ ② r♥ t t♦♠♥s♦♥ ♦s♦ stts♣ t♦ s♥ ♠♥s♦♥ ♥♥ t ♦s♦ ♥t♦♥s r♦♠ V z,f

t,j (aj, hoj)

t♦ V z,ft,j (yj) r stt♦♥r② ♥t ♦rt yj s t s♠ ♦ t stt♦♥r② st♦ ♦ ♥♥

ssts aj ♥ ♦♥r♦♣ ♦s♥ srs hoj s♠t♥♦s② ①♣♥ t st ♦

♦s♦ ♦ rs r♦♠ aj+1, hoj+1; xj, nj, c

ij, c

gj , h

rj t♦ yj+1; xj, nj, c

ij, c

gj , h

rj , h

oj , aj

♦♦s♥ t ♦♣t♠ ♦ tr ♥t ♦rt ♥r ♣rt ♦rst ♦s♦s ♥♦t rt♥t② ts ♦♠♣♦st♦♥ r♦ss ♥♥ ssts ♥ ♦♥r♦♣ ♦s♥ srs ② ♥t♦♥ ♦ t ♥t♦♥ ❲t ts ♥♦r♠t♦♥ t② ♠② t♥ ♦♦st ♦♣t♠ ♦♠♣♦st♦♥ ♦ rr♥t ♥t ♦rt r♦ss ts s♠ ♦ rs ♥ t♠♦♥sst♥t s♦♥ ♦♥ t rrs ②♥♠ ♣r♦r♠ ② rs ♥t♦♥s sr ♥ ♣♣♥① rts ts strtr

♥ ♣rs qsts ♥ ♦r♥♠♥t ♣♦② s ♥ ♦s♦s ♦♠♣r t♥rt tt② ♥rt ② s♦♥s ss♦t t ♦♥♥ ♦♠ ♥st t r♥rt ② s♦♥s ss♦t t r♥t♥

Vz,ft,j (yj) = maxV o;z,f

t,j ,Vr;z,ft,j

r Vo;z,ft,j ♥ V

r;z,ft,j r t rr♥t ♣r♦ ♥t♦♥s ss♦t t ♦♥♥

♥ r♥t♥ rs♣t② ♦r ♦s♦ ♦ ♠♦r♣ (j, z, f)

♥ t ♦r t sss♦♥ ♦ tr♥ stt♦♥rt②

♥t♦♥ ♦r ♦s♦ tt ♦♠♦♥r ♥ ♣r♦ t s

Vo;z,ft,j (yj) =

maxyj+1;xj ,nj∈N

Uo;z,st,j (xj, nj) + βπjV

z,st+1,j+1(yj+1) f = s

maxyj+1;xj ,n1

j ,n2j∈N

Uo;z,mt,j (xj, n

1j , n

2j) + βπjV

z,mt+1,j+1(yj+1) f = m

r

Uo;z,ft,j ≡

maxcij ,c

gj ,h

oj

log(xj)− ψs n1+ζs

j

1+ζs− F s f = s

maxcij ,c

gj ,h

oj

log(xj)− ψm,1 (n1j )

1+ζm,1

1+ζm,1 − ψm,2 (n2j )

1+ζm,2

1+ζm,2 − Fm f = m

xj ≡(

σcηj + (1− σ)(hoj)η)1/η

s sst ♦ t st ♦ ♦♥str♥ts ♦r ts ♦♣t♠③t♦♥ ♣r♦♠ t ♦s♦♦♥sr♥ ♥ ♦♠♦♥r ♥ ♣r♦ t s

cMj + cgj +(rpt + δo) hoj + yj+1ΥA ≤ (1 + rpt ) yj + ˜beqt+ iz,ft,j −T z,f

t,j − κz,fj −φrhrj+1ΥA

hoj ≥ o

yj ≥ γhoj

hrj = 0

♥t♦♥ ♦r ♦s♦ tt r♥tr ts t ♦r♠

Vr;z,ft,j (yj) =

maxyj+1;xj ,nj∈N

Ur;z,st,j (xj, nj) + βπjV

z,st+1,j+1(yj+1) f = s

maxyj+1;xj ,n1

j ,n2j∈N

Ur;z,mt,j (xj, n

1j , n

2j) + βπjV

z,mt+1,j+1(yj+1) f = m

r

Ur;z,ft,j ≡

maxcij ,c

gj ,h

rj

log(xj)− ψs n1+ζs

j

1+ζs− F s f = s

maxcij ,c

gj ,h

rj

log(xj)− ψm,1 (n1j )

1+ζm,1

1+ζm,1 − ψm,2 (n2j )

1+ζm,2

1+ζm,2 − Fm f = m

xj ≡(

σcηj + (1− σ)(hrj)η)1/η

s sst ♦ t st ♦ ♦♥str♥ts ♦r ts ♦♣t♠③t♦♥ ♣r♦♠ t ♦s♦♦♥sr♥ ♥ r♥tr ♥ ♣r♦ t s

cMj + cgj + prt hrj + yj+1ΥA ≤ (1 + rpt ) yj + ˜beqt + iz,ft,j − T z,f

t,j − φohoj+1ΥA − κz,fj

hrj ≥ r

yj ≥ ②z,f

hoj = 0

rss ♦ rs♥t stts ♦s♦s t ♦♦♥ qt② ♥ ♥qt② ♦♥str♥ts ♠♣♦②♠♥t ① ♥ r ♦sts ♥ ♥t ♥ tr♠♥ ♦♥t♦♥s

yj ≡ hoj + aj

cj ≡ (cij)θz,f (cgj )

(1−θz,f )

nǫj =

1− lǫj − nhj (n

ǫj) ∀j ≤ R

0 ∀j > Rǫ = 1, 2

iz,ft,j ≡

njwt③z,sj + ssz,sj f = s

(n1j + µzn2

j)wt③z,mj + ssz,mj f = m

κz,fj =

ccz,sνz,sj nj f = s

ccz,mνz,mj n2j f = m

F s =

φs nj > 0

0 nj = 0

Fm =

φm n2j > 0

0 n2j = 0

cij ≡

cMj + chj (nhj ) f = s

cMj + ch,2j (nh,1j ) + ch,1j (nh,2

j ) f = m

T z,ft,j ≡ ˜tax

z,ft,j + τat r

ptˆaz,ft,j + τ prt

ˆiz,ft,j − ( ˜trwz,ft,j + ˜trlt − ˜lstt) + (τ slt

ˆiz,ft,j + τ slpt hot,j)

cij ≥ i

cj = cij cij = i

y1 = a1

ho1 = yJ+1 = 0

Vz,ft,J+1 = 0

qr♠

♦r ♦s♦ ♦♦rt j ♣r♦tt② t②♣ z ♥ ♠② ♦♠♣♦st♦♥ f ♦s♦s ♦r♥r② ♦♥s♠♣t♦♥ ci rt ♥ cg ♠rt ♦r ♦rs n n1 ♥n2 ♦♥r♦♣ ♦s♥ srs ♦♥s♠♣t♦♥ ho r♥t ♦s♥ srs ♦♥s♠♣t♦♥ hr ♥ tr ♥t ♦rt y′ s ♦♥tr♦ rs ♦s♦s rr♥t ♥t ♦rty s tr ♥♦♥♦s ♥ stt r ♥ tr ♣r♦tt② t②♣ s ♠② ♦♠♣♦st♦♥ s tr ①♦♥♦s stt rs ♥♦♥♦s rt stt rsr t ♠rt ♦r s♣♣② N ♦♥r♦♣ ♦s♥ ♣t Ho r♥t ♦s♥♣t Hr ♣♦sts D ♣rt s♥ss ♣t K ♣ ♣t G r ♦r♥♠♥tt B ♥ r stt ♥ ♦ t① ♥str♠♥ts ♥ tr♥sr ♣②♠♥ts ss♦tt ♥ t① s②st♠ t st ♦ r ♥♦t ② T ♥

♥t♦♥ ♣rt♦rst stt♦♥r② rrs qr♠ s ♦♠♣rs

♦ ♠sr ♦ ♦s♦s Ωz,ft,j ♥t♦♥ V z,f

t,j (y) ♦t♦♥ ♦ ♦s♦

s♦♥ rs ci;z,ft,j (y) cg;z,ft,j (y) nz,st,j (y) n

z,m,1t,j (y) nz,m,2

t,j (y) ho;z,ft,j (y) hr;z,ft,j (y) yz,ft+1,j+1(y)

az,ft,j (y) ch;z,ft,j (y) ♣rs wt, rt, p

rt , ρt, r

pt rts Nt, H

ot , H

rt , Dt, Kt, Gt, Bt; Ct, It, Gt

♥ t st ♦ t① ♥str♠♥ts ♥ tr♥srs T ss♦t t ♥ t① s②st♠ s tt

♦s♦ s♦♥ rs r t s♦t♦♥s t♦ t ♦♥str♥ ♦♣t♠③t♦♥ ♣r♦♠

♥ qt♦♥s

r♦♦♥♦♠ rts r ♦♥sst♥t t ♦s♦ ♦r s tt

Nt =

∫

Z

∫

J

Ωz,st,j ③

z,sj nz,s

t,j (y) + Ωz,mt,j ③

z,mj

(

nz,1t,j (y) + nz,2

t,j (y))

dj dz

Hot =

∫

Z

∫

J

∑

f=s,m

Ωz,ft,j h

o;z,ft,j (y) dj dz

Hrt =

∫

Z

∫

J

∑

f=s,m

Ωz,ft,j h

r;z,ft,j (y) dj dz

Dt =

∫

Z

∫

J

∑

f=s,m

Ωz,ft,j a

z,ft,j (y) dj dz

Ct =

∫

Z

∫

J

∑

f=s,m

Ωz,ft,j

(

(ci;z,ft,j (y)− ch;z,ft,j (y)) + cg;z,ft,j (y) + κz,ft,j

)

dj dz + ceolt

r♠s r ♣rt t♦rs ♦ ♣r♦t♦♥ t♦ ♣r♦ ♦t♣t t ♣r♦t♠①♠③♥ s

t♥ ♣ ♣t ♥ t♦r ♣rs ♦♥sst♥t t t ♦♦♥ s ♥

wt = (1− α− g)Ggt K

αt N

−α−gt

rt = K−1t

(

(1− τ bust )(1− τ slbt )(α + g)(Ggt K

αt (AtNt)

1−α−g) + τ bust˜lsd

bus

t

)

sst ♠rt rs s tt

Dt+1 = Kt+1 + Hrt (ΥPΥA)

−1 + Bt+1

r t ♥♥ ♥tr♠r② ♦♣t♠② ♦ts ♣♦sts ♥t♦ ♣r♦t ♣rt

♣t ♥ r♥t ♦s♥ s♦ tt t ♦♦♥ ♥♦rtr ♦♥t♦♥ ♦s

prt = rt+1 − δK + δr −

(

∂Ξt

∂Kt+1

+∂Ξt+1

∂Kt+1

)

♥ s ♥ t♦ ♣t ❵ssst ♣r♥ ♦ r ♦r♥♠♥t ♦♥s s♦ tt

ρt = rt −

rtr♠♦r t rt ♦ rtr♥ ♣ t♦ ♦s♦s ♦♥ ♣♦sts s tr♠♥ ②

♣♣t♦♥ ♦ ③r♦ ♣r♦t ♦♥t♦♥ s♦ tt

rpt = D−1t

(

(rt − δK)Kt − Ξt + ptHrt − δrHr

t−1(ΥPΥA)−1 + ρtBt

)

♦♦s ♠rt rs s tt

F (G, Kt, Nt) = Ct + It + Gt

r ♣rt rt ♥st♠♥t s ♥ s

It ≡ IKt + Iot + Irt + ΦHt

t

IKt = Kt+1(ΥPΥA)− (1− δK)Kt + Ξt

Iot = Hot+1(ΥPΥA)− (1− δo)Ho

t

Irt = Hrt − (1− δr)Hr

t−1(ΥPΥA)−1

ΦHt =

∫

Z

∫

J

∑

f=s,m

Ωz,ft,j

(

φoho;z,ft+1,j+1(y) + φrhr;z,ft+1,j+1(y))

dj dz

♥ r rt ♦r♥♠♥t ①♣♥trs s ♥ s

Gt ≡ Cfedt + Csl

t + Ifedt + Islt

t

Ifedt = Gfedt+1(ΥPΥA)− (1− δg)Gsl

t

Islt = Gslt+1(ΥPΥA)− (1− δg)Gsl

t

r ♦r♥♠♥ts t ♦♦s t ♦ ♠♦t♦♥

Bt+1(ΥPΥA) = Cfedt + Ifedt − (T hh

t + T bust + T beq

t ) + (1 + ρt)Bt

♥ ♠♥t♥s s② sst♥ ♣t s♦ tt

limk→∞

Bt+k∏k−1

s=0(1 + ρt+s)= 0

r ♥t r t① r♣ts r♦♠ ♦s♦s r♠s ♥ qsts r

T hht =

∫

Z

∫

J

∑

f=s,m

Ωz,ft,j

(

˜taxz,ft,j +τ

at r

ptˆaz,ft,j +τ

prtˆiz,ft,j −( ˜trw

z,ft,j +

˜trlt− ˜lstt)−ssz,ft,j

)

dj dz

T bust = τ bust

(

(1− τ slbt )(Yt − wtNt)− ˜lsdbus

t

)

T beqt = τ beqt (1− Λ) (ΥA)

∫

Z

∫

J

(1− πj)∑

f=s,m

Ωz,ft,j yt+1,j+1 dj dz

stt ♥ ♦ ♦♠♣♦st ♦r♥♠♥t ♠♥t♥s ♥ t

T slht + T slb

t = Cslt + Islt

r ♥t stt ♥ ♦ t① r♣ts r♦♠ ♦s♦s ♥ r♠s r

T slht =

∫

Z

∫

J

∑

f=s,m

Ωz,ft,j (τ

sltˆiz,ft,j + τ slpt hot,j) dj dz

T slbt = τ slbt

(

Yt − wtNt

)

♠sr ♦ ♦s♦s s t♠♥r♥t

Ωz,ft+1,j = Ωz,f

t,j

♥t ♦rt ♦ ♦s♦s tt ♦r r♥ t ♠①♠♠ J s ♦t

t♦ ♥♦ ♦♥s♠♣t♦♥ ①♣♥trs t♦ qsts ♠♦♥ t ♥ s tt

ceolt = Λ (ΥA)

∫

Z

∫

J

(1− πj)∑

f=s,m

Ωz,ft,j yt+1,j+1 dj dz

˜beqt = (1− τ beqt )(1− Λ) (ΥA)

∫

Z

∫

J

(1− πj)∑

f=s,m

Ωz,ft,j yt+1,j+1 dj dz

♥t♦♥ st②stt ♣rt♦rst stt♦♥r② rrs qr♠

s ♣rt♦rst stt♦♥r② rrs qr♠ r r② r♦tst r

t r s t♠ ♥r♥t

sr♣t♦♥ ♦ ♦t♦♥ ♦rt♠

t② tt

♥ t st ♦ t① ♥str♠♥ts ♥ tr♥srs T ♥ strt♥ ss ♦r ♥t qsts ˜beqt ♠ ss ♦r t st ♦ ♥♦♥♦s rt stt rs Nt, H

ot , H

rt , Dt, Kt, Gt, Bt ♥ s ts sss t♦ ♦♠♣t t ♦ ♣rs

wt, rt, prt , ρt, r

pt

♦r ♦s♦ ♠② ♦♠♣♦st♦♥ f = s,m ♥ ♦r ♣r♦tt② t②♣ z ∈ Z♦t♥ t ♦♣t♠ ♦s♦ s♦♥ rs ss♦t t t ♥t♦♥ ♥qt♦♥ t j ∈ J ② rs rrs trt♦♥s st t♦qt♦♥s

♥♥♥ t t ♠①♠♠ J ♥②t② ♦♠♣t t ♦♣t♠ ♦s♦ ♦s♥ ♦r♥r② ♦♥s♠♣t♦♥ ♥ rt ♥ ♥ tr♠s ♦ r ♦srt s ♦r rr♥t ♥t ♦rt yJ ♦t ♦r t ♦♣t♠③t♦♥ ♣r♦♠♣rtr t♦ rr♥t ♣r♦ r♥trs ♥ ♦♠♦♥rs s sr ♥ t♦♥ ❯s ts ♦s t♦ ♦♠♣t t ♥t♦♥s ss♦t t r♥trs♥ ♦♠♦♥rs rs♣t② stt♥ t ♥t♦♥ t♦ r ♥t♥♠r ♦r t♦s ♦s tt ♦t ♥qt② ♦♥str♥ts ♦r ♥t♦rt ♥♦ st V

z,ft,J (yJ) q t♦ t ♠①♠♠ ♦ V o;z,f

t,J ♥ Vr;z,ft,J t♦r

t ♦s♦ ♦s ss♦t t t ♦♣t♠ ♦s♥ stts t ♥t♦rt ♥♦

♦r s j ∈ J − 1, . . . , 1 r♣t ♣rt ♦r r② ♣♦ss (yj, yj+1) ♦♠♥t♦♥ ♦ ♥t ♦rt ♥♦s t♦ ♦t♥ t ♦♣t♠ ♦s♦ ♦s

❯s t ♦♣t♠ ♦s♦ s♦♥ rs ♦t♥ r♦♠ t ♣r♦s st♣ t♦ s♠t② ♦s ♦r ♠♦r♣ t t ♥t ♦♥t♦♥s ♦r ♥t ♦rt ♦y1 = a1 ♥ ♦♥r♦♣ ♦s♥ ho1 = 0 ♠t♦♥ ♥s ne ♥♦♠♥ts r♦♠ ♠♦r♣ ♦r t♦t ♦ nf ×nz×J ×ne s♠t♦♥s ♦♠♣t rts ♥ ♠♣ ♣rs

♦♠♣r t ♥ st ♦ rts Nt, Hot , H

rt , Dt, Kt, Gt, Bt ♥ ♥t

qsts ˜beqt t♦ t ♥t sss t ♦ ts ♥ rts r s♥t②♦s t♦ t ss s t♥ st②stt qr♠ s ♦t♥ ♥ t ♣r♦r♠ ♥ tr♠♥t ♥♦t ♣t ss ② t♥ ♥r ♦♠♥t♦♥♦ t ♦r♥ ss ♥ t ♥ ♦t♥ r♦♠ ♣♣t♦♥ ♦ qr♠♦♥t♦♥s ❯s ts ♥ sss t♦ ♦♠♣t ♥ s ♦r t st ♦ ♣rswt, rt, p

rt , ρt, r

pt tr♥ t♦ st♣

♥ t ♠♦ s s♦ rs ♦♣t♠ ♦s ♦r ♦s♦ j + 1 r ♥♦♥ ♦rt♦s ♦ ♦s♦ j r♦r ho

t,j+1 ♥ cgt,j+1

r s s ♣r♦①s ♦r hot,j ♥ c

gt,j ♦r ♣r♣♦ss ♦

♦♠♣t♥ t♠③ t① t♦♥s r♥ t ♦♣t♠③t♦♥ st♣ s ♦s ♦r sst♥t rt♦♥♥ ♦♠♣tt♦♥ r♥ ② ♦♥ sr ♦r s♦♠ s♣ s♣♣♦rt ♦r ho ♥ cg ♥ t♦♥ t♦ tsr ♥ ♦r ♦r s♣♣② ♥ ♦rr t♦ ♥♦ i

z,ft,j ♦r t ♦r♥ ♣♦♣t♦♥ ♣♣r♦①♠t♦♥

rr♦r ♥tr♦ ② ts ♣r♦①② s ♠t ♦t t sr ♦ ♦s♦s t♠③ t① t♦♥s ♥t ② s♠♦♦t ♣t ♦ ts rs

r♥st♦♥ Pt

♦♦s t ♥♠r ♦ tr♥st♦♥ ♣r♦s tp s♥t② r s♦ tt t ♦♥♦♠②rs st② stt ♦♦♥ ♣♦② ♥

♦♠♣t t ♥t st② stt ♦r t ♣♦② s♥ ♥ t ♥ st② stt♥r t ♥ ♣♦②

Pr♦ ♥ ♥t ss ♦r t t♠ ♣t ♦ t st ♦ rt rs

♦♠♣t t tr♥st♦♥ ♣t

♦r ②rs ♦r ♣♦② ♥ s s♦♥ rs r♦♠ st② stt

trt♥ t rst ②r ♦ ♣♦② ♥ ♦♠♣t ♥ s♦♥ rs ♦r ♥ts♦ ♠♦r♣ ♦r t ♣r sss ♦r tt t♠ ♣r♦

♥ s♠t♦♥ strt♥ J ②rs ♦r ♣♦② ♥ s♦ tt ② t rst tr♥st♦♥ ②r tr r J s ❯s ♥ s♦♥ rs ♥ ♣r sss t♦s♠t ♦rr tp ♣r♦s

♦♠♣t ♥ rts ♥ ♣rs ♦r t♠ ♣r♦

♦♠♣r t ♥t ss ♦r t t♠ ♣t ♦ t rt rs t♦ tr ♥t♠ ♣t ♦t♥ r♦♠ t ♣r♦s st♣ t♠ ♣ts r s♥t② ♦str♠♥t t ♣r♦r♠ ♥♦t ♣t t sss ② t♥ ♥r ♦♠♥t♦♥♦ t ♥ t♠ ♣t ♥ t ♦ t♠ ♣t tr♥ t♦ t♣

s

rt ♥ s♥ t ♠♣♦②♠♥t tts ② ②♣ ♦ ❲♦rr

②♣ ♦ ❲♦rr t ♦ ♦

P ❯ P ❯ P ❯

♥ rr Pr♠r② rr ♦♥r②

♦ts ♠② ♥♦t s♠ t♦ t♦ r♦♥♥

rt ♥ s♥ t rt t♦s

rt t♦ t ♦ ♦

♦♠♦♥rs♣ rt♦

Prt s♥ss ♥st♠♥t t♦t♦t ♣rt ♥st♠♥t rt♦

Prt s♥ss ♥st♠♥t t♦♦t♣t rt♦

t ①♦♥♦s Pr♠trs

♠♦r♣s

r♠♥ s R J t ♦ ♣♦♣t♦♥ r♦t υP

Pr♦t♦♥

t ♦ t♥♦♦ ♣r♦rss υA Prt ♣t sr ♦ ♦t♣t α P ♣t sr ♦ ♦t♣t g Prt ♣t ♣rt♦♥ rt δK Prt ♣t st♠♥t ♦st ♣r♠tr ξK

♦s♥

♥r♦♣ ♦s♥ ♠♥♠♠ ♦♥♣②♠♥t γ ♦s♥ stts st♠♥t ♦st φo φr ♦s♥ srs ♣rt♦♥ rt δo δr ♥r♦♣ ♦s♥ ♠♥♠♠ o ♥r♦♣ ♦s♥ ♠♥♠♠ o

Prr♥s

t s♦♥t t♦r β ♦♥♦s♥ ♦♥s♠♣t♦♥ sr ♦ ♦♠♣♦st σ ♦s♥♥♦♥♦s♥ ♦♥s♠♣t♦♥ ssttt♦♥ stt② η ❯tt② rtr ♣r♠tr ζf,ǫ ♥t♥s ♦r ♠r♥ stt② ψsψm,1ψm,2 395.1, 264.6, 176.7♥t♥s ♦r ♠r♥ stt② ψsψm,1ψm,2 396.3, 279.9, 177.6①t♥s ♦r ♠r♥ ① ♦st φs φm 0.354, 0.155①t♥s ♦r ♠r♥ ① ♦st φs φm 0.393, 0.151♦r♥♠♥t

P ♣t ♣rt♦♥ rt δg

r♥ tt♦♥r② r♥s♦r♠t♦♥s ♦r t ❱rs

rt ❱rs

C it ≡

C it

AtPt

Hot ≡

Hot

AtPt

Hrt ≡

Hrt

AtPt

Dt ≡Dt

AtPt

Kt ≡Kt

AtPt

Cgt ≡

Cgt

AtPt

Bt ≡Bt

AtPt

Gt ≡Gt

AtPt

Tt ≡TtAtPt

Nt ≡Nt

Pt

♥ ❱rs

cit ≡ctAt

hot ≡hotAt

hrt ≡hrtAt

cgt ≡cgtAt

at ≡atAt

yt ≡ytAt

it ≡itAt

Tt ≡Tt

At

˜trst ≡trstAt

sst ≡sstAt

˜taxt ≡taxtAt

˜beqt ≡beqtAt

wt ≡wt

At

Ωt ≡Ωt

Pt