pe pt dt = e pt Probabilty of death given survival till t : pe pt = p Expected life at t : pe(s t)p ds = e (s t)p t =

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1 BLANCHARD Probabiliy of Deah: π () = pe p ; Probabily of living ill : Ω () = pe p d = e p Probabily of deah given survival ill : pe p = p e p Expeced life a : (s ) pe (s )p ds = p 1 Populaion normalized o consan cohor size p. Cohor born a 0 has size pe p : (prob of living ill imes pop. size-deerminisic) Populaion: pe(s )p ds = e (s )p = 1 Insurance exiss. Agens die wih prob. densiy p. So no ne profis for insurance implies hey receive p for each uni hey leave. They plan o leave all, and receive pw if alive. On average insurance companies pay ou pw and receive pw from esaes where W is aggregae wealh. 1

2 Max ST dw (s, ) d 0 = lim e v ln c (v, s) e (θ+p)( v) dv = (r () + p) w (s, ) + y (s, ) c (s, ) v (r(u)+p)du w (s, v) dv Noes: If r (u) is consan v (r (u) + p) du = (r + p) (v ). To assure zero erminal wealh r (u) > p is assumed. Also Ω (v ) = e p( v) in Yaari s erms. r > p is assumed. Hamilonian: FOC H = ln c + λ ((r + p) w + y c) c 1 ċ = λ; c = λ λ λ = λ ( r p + p + θ) = λ (r θ) ċ = c (r θ) 2

3 Budge c (s, v) e v (r(u)+p)du dv = w (s, ) + h (s, ) (r + p) (w + h) c = ẇ + ḣ Human capial is discouned fuure labor income: h = ḣ = (r + p)h y y (s, v) e v (r(u)+p)du dv ẇ = (r + p) (w + h) c ḣ ẇ = (r + p) w + y c 3

4 Conjecure c (s, ) = (p + θ) (w + h) ċ = (p + θ) ((r + p) w + y c + (r + p)h y) ċ = (p + θ) ((r + p) (w + h) c) ċ = (r + p) (p + θ) (w + h) (p + θ) c ċ = (r θ) c checks ou 4

5 AGGREGATE CONSUMPTION Aggregae H H () = C = (p + θ) (W + H) pe p(s ) y (s, v) e v (r(u)+p)du dvds Assume common income or wage y(v), and noing ha populaion inegraes o 1: H () = = = pe p(s ) [ y (s, v) e v (r(u)+p)du dvds ] y (s, v) e v (r(u)+p)du pe p(s ) ds [Y (v) e v (r(u)+p)du ds Ḣ = (r + p) H Y () ] dv dv 5

6 Aggregae W W () = Ẇ = w (, ) pw + = pw + w(s, )e p(s ) ds dw (s, ) pe p(s ) ds d ((r + p) w + y c) pe p(s ) ds = pw + (r + p) W + Y C = rw + Y C So while hose alive have heir wealh grow a (r + p), aggregae wealh accumulaes a rae r, since pw is a ransfer from he dead. 6

7 Aggregae Dynamics C = (p + θ) (H + W ) Ċ = (p + θ) ( Ḣ + Ẇ ) Ċ = (p + θ) (rw + Y C + (r + p) H Y ) = (p + θ) ((r + p) (W + H) C pw ) = (r + p)c (p + θ) C (p + θ) pw = (r θ) C (p + θ) pw Ċ = (r θ) C (p + θ) pw Ẇ = rw + Y C Seady Sae C = (p + θ) pw r θ 7

8 OPEN ECONOMY Ċ = (r θ) C (p + θ) pf F = rf + Y C [ ] r θ (p + θ) p J = 1 r De: (r θ) r (p + θ) p < 0? A Seady Sae: ( ) (p + θ) p C = rf (r θ) r Noe: r θ. So F may be negaive. If r is oo big however he economy may explode: you need r < p + θ, or r θ < p. This implies r (r θ) < p (p + θ). So De < 0. Furhermore, ) If (r θ) > 0 his implies > 1, or C > rf. ( (p+θ)p (r θ)r 8

9 CLOSED ECONOMY: F (K) = F (K, 1) δk, r = F Ċ = (r (K) θ) C (p + θ) pk K = F (K) C J = For C > 0,a seady sae, r (k ) > θ. r θ (p + θ) p 1 F If r θ > p, (r θ) C = (r θ) F (K) > pc (r θ) F (K) = (p + θ) pk > pc = pf (K) (p + θ) K > F (K) The laer implies, if r > p + θ, rk > F (K),bu oupu mus exceed MP K K, so r (K) < (p + θ). So De J < 0. 9

10 Declining Labor Income: y(s, v) = ay (v)e α(s v) Toal income a :sum of cohor income weighed by pop.size v Y (v) = y(s, v)pe p(s v) ds = Y (v) ap p + α a = p + α p h (s, ) = ay (v) e α(s v) e v = e α(s ) ( p + α p ) (r(u)+p)du dv Y (v) e v (r(u)+p+α)du dv 10

11 = = H () pe p(s ) ( ( ) p + α e α(s ) p ( (p + α) e (p+α)(s ) = e (p+α)(s ) ( = Dynamics: Y (v) e v (r(u)+p+α)du dv ( Y (v) e v (r(u)+p+α)1 du dv C = (θ + p) (H + W ) Ḣ = (r + p + α) H Y Ẇ = rw + Y C Compue Ċ using Ẇ, ) Y (v) e v (r(u)+p+α)du dv ds ) ds Y (v) e v (r(u)+p+α)du dv ) Ḣ, and subsiue ) 11

12 for H from firs equaion above: Ċ = (r (k) + α θ) C (p + α) (p + θ) K K = F (K) C Seady Sae: (p + α) (p + θ) K C = (r (K) + α θ) (p + α) (p + θ) F (K) = r (r (K) + α θ) rk Define ˆk r ) (ˆk = θ α :Noe ha θ α ) can be negaive, so r (ˆk < 0. Plo seady sae C as a funcion of K, which asympoes a [ ˆk. (r (k) + α θ) r J = (K) C (p + α) (p + θ) 1 F (K) DET = r (r (k) + α θ)+r (K) C (p + α) (p + θ) I is negaive as before because we assume (r (k) + α θ) < (p + α) (p + θ) and r (K) < 0. ] 12

13 If we allow Money: Le m = M P, W = K + m ṁ m = σ π where σ,is he growh of nominal balances, π is inflaion. Equilibrium condiion: reurn on money equals reurn on capial, r (k). If σ = 0, r (k) = π, he reurn on holding money equals o he deflaion rae (Compare wih he case below where money eners uiliy, and has an addiional uiliy reurn.) If money pays ineres σ, in proporion o money holdings, hen in equilibrium, r (k) = σ π. If σ = r (k), in equilibrium, π = 0. In any case, now equilibrium condiions are: Ċ = (r (K) θ) C (p + θ) p (K + m) K = F (K) C ṁ = r (k) m Seady saes: m = 0, r (k) = 0 CRRA UTILITY = U (c) = (1 σ) 1 c (1 σ) dc (s, ) = σ 1 (r () θ) c (s, ) d c (s, ) = 1 (w (s, ) + h (s, )) e σ 1 v [(1 σ)(r(u)+p) (θ+p)]du dv 13

14 dc (s, ) ) = (ẇ 1 + d ḣ 2 (w + h) Ċ = 1 ((r + p) (H + W ) pw C) 1 (W + H) = 1 (r + p) C 1 C 1 pw 1 (W + H) ( ) (r + p) = C σ 1 pw 1 ((r + p) (1 σ) (θ + p)) So Ċ = σ 1 (r () θ) C pw 1 K = F (K) = 1 σ 1 ((1 σ) (r (k) + p) (θ + p)) Noe: Solve for seady sae as a funcion of r and plug ino ( Ċ = 0, ) (σ 1) (r + p) + (θ + p) C = pk r θ as opposed o log uiliy, where C = cases dc dk according o σ 1. If σ < 1, dc dk 14 ( ) p(p+θ) r θ pk.consider

15 may be negaive: increasing K causes decreasing r.denominaor will decrease if r increases, as will numeraor, bu denominaor can dominae. Local Dynamics: Cr σ (p + α) 1 (p + α) k 2 1 F 0 r θ+α σ 0 (1 σ) σ r (σ 1)(r+p+α)+(θ+p) σ DET < 0, T RACE > 0, ROOT ST RUCT URE : + + INTRODUCING MONEY IN UTILITY Max 0 (ln c + ln m) e (θ+p) d ẇ = (r + p) w + y c (π + r) m H = ln c + ln m +λ ((r + p) w + y c (π + r) m) FOC c 1 1 = λ, = λ (π + r), c = (π + r) m m ċ = c (r θ) Wealh 15

16 w + h = h = Differeniaing h, w + h : (c (s, v)) e v (r+p)du dv (y (s, v) m (r + π)) e v (r+p)du dv ḣ = y (s, v) + m (r + π) + (r + p)h c + (r + p) w = ẇ + ḣ = ẇ c y + (π + r) m ẇ = (r + p) w + y c (π + r) m Consumpion funcion: c = (θ + p) (w + h) ċ = ((θ + p) [(r + p) w + y c (π + r) m y + m (r + = [(r + p) (θ + p) (w + h) (θ + p) c] = (r + p) c (θ + p) c = (r θ) c 16

17 Aggregaing H = W = (Y (v) (π + r) M (v)) e v (r+p)du dv w (s, ) e p(s ) ds Ḣ = (r + p) H Y + (r + π) M inegraing by pars dw (s, ) Ẇ = w (, ) pw + pe p(s ) ds d = pw + (r + p) W + Y C (π + r) M C = (θ + p) ( (W + H) (r + p) H Y + (r + π) M Ċ = (θ + p) pw + (r + p) W + Y C (π + r) M Ċ = (θ + p) ((r + p) (W + H) C pw ) Ċ = (r + p) (θ + p) (W + H) (θ + p) C p (θ + p) W Ċ = (r + p) C (θ + p) C p (θ + p) W = (r θ) C (θ + p) pw 17

18 From FOC: (π + r) M = C (π + r) M = (θ + p) (W + H) πm = C rm M = πm = C + rm Dynamics M = πm = C + rm K = F (K) C Ċ = (r θ) C (p) (p + θ) (K + M) INTRODUCING ENDOGENOUS LABOR ( )) Max ln c + aln ( l l e -(θ+p) FOC 0 ẇ = (r + p) w + ly + c a l l = λy, ly ac = yl ċ (r θ) c 18

19 (p + α) h = e ( ly ) α(s ) ca e v (r+p+α)du dv p (p + α) = e α(s ) (ly) e v (r+p+α)du dv p ( ) (θ + p) c = (w + h) ( 1 + a ) (θ + p) ((r ċ = + p) w + ly + c + (r + p + α) h y l) ( 1 + a ) (θ + p) ċ = ((r + p) w + c + (r + p + α) h + ac) 1 + a = c (r θ) ( ) (θ + p) C = (W + H) 1 + a ( ) ( (θ + p) rw + LY C Ċ = 1 + a + (r + p + α) H LY (p + θ) (p + α) Ċ = (r θ) C+ W 1 + a ) 19

20 Dynamics (p + θ) (p + α) Ċ = (r θ) C + K 1 + a K = F (K, L) C L = LY BC Y Y = F L (K, L) 20