Essays in Macroeconomics

Università degli Studi di Sassari

Essays in Macroeconomics

Direttore della scuola di Dottorato:

Prof. Michele M. Comenale Pinto

Tutor:

Prof. Gerardo Marletto

Tesi di Dottorato di Ricerca di:

Marco Delogu

La presente tesi è stata prodotta durante la frequenza del corso di dottorato in Diritto ed Economia dei sistemi produttivi dell’Università degli Studi di Sassari, a.a. 2011/2012 - XXVII ciclo.

Ht Lf,t Yf,t= Af,tHt↵L1 ↵f,t ↵ Af,t t kt ⇣ zf,t=_LH_f,tt ⌘ Af,t= A0zf,tkt" A0 " wh,t= (1 ⌧ ) ↵A0zf,t↵+ 1k"t wl,t= (1 ⌧ ) (1 ↵) A0zf,t↵+ kt"

Yi,t= BLi,t Li,t B A0 B = A0 B wl,t B 1, Yi,t 0 ⇣ wl,t B 1 ⌘ Yi,t= 0 • B • Nt⌘ Ht+ (Lf,t+ Li,t) Yi,t= BLi,tk"t " < "

8 < : zt=_Lf,tH_+Lt _i,t zf,t=_LH_f,tt {h, l} • nh, nl • q • ¯m, m ⌧

kt+1= ⌧ Yf,t Nt+1 wl,i,t> (1 ⌧ ) wl,f,t ! kt< k (zf,t) (↵ + < 1)

zf,t= 8 < : zf,t⌘ zt zf,t = ⇣ (1 ⌧ )(1 ↵)k " t ⌘ 1 ↵+ (zf,t, kt) (zt> zf,t) • zt+1= Ht+1 Lt+1 = (1 m) (nz¯ t+ q) (1 q) (1 m) • n nh nl

gt= 1 + zt = nh(1 m) z¯ t+ nl((1 m) (1 q) + (1 m) q)¯ • kt+1= ⌧ A0z +↵ t kt" • kt+1=⌧ A0zf,t ( +↵ 1)_k" tzt Ht= ¯Ht Lf,t+ Li,t= ¯Lt N0 H0 k0 {↵, , ", ⌧, nh, nl, q, ¯m, m} {wh,t, wl,t} {zt} {zf,t} {kt}

• • {wh,t, wl,t} H¯t, ¯Lt • • • t t {z0, k0} {zt, kt}1t=0 zt= zt+1 kt= kt+1

• dzss d ¯m < 0 • dzss dm > 0 • dzss dq > 0 1 m 1 m¯ > n • =_{(1 ↵)(1 ⌧ )} • ⇥ = nl((1 m) (1 q) + (1 m) q)¯ • zss nh(1 m) (z¯ ss) " (↵+ ) " 1"" + ⇥ (z⇤) 1 " " ⌧ A 0 0

nh(1 m) (z¯ ss)

" (↵+ )

" 1"" + ⇥ (z⇤) ↵+

t t ⌧ < " (↵ + ) (" + + ↵) t + 1 t

µ

Pha= [ln (wh,t) < ln (w⇤t) + ln (1 xt) + ¯⌘t,i]

i t

F (⌘) = e e(⌘µ ) "✏< Pha= e 1 µln(w⇤t) eµ1ln(w⇤t)_{+ e}µ1ln((1 ⌧ )↵A0z↵+f,t 1k " t) ¯ mt= w⇤ t w⇤ t + (1 ⌧ ) ↵A0zf,t↵+ 1k"t µ

zt = zt+1 zt zt+1 zt N0 H0 k0 {µ, ↵, , ", ⌧, nh, nl, q, m} {wh,t, wl,t, wt⇤} {zt} {zf,t} {kt} { ¯mt} • • • (zt= zt+1) (kt= kt+1)

•

(↵ + ) < 1

[0 1] (↵ + ) < 1

zt< q

(1 q) (1 m) n

• (⌧, ", , n, µ)

n

µ

nl zf,00,j = H00,j (Lf,00,j+ Li,00,j) s98,j(L00,j+ H00,j) s98 ↵

M R00,j DY00,J wr00,j = (1 + M R00,j)DY00,j ↵ ↵j= wrj,00zj,00,f 1 + (wrj,00zj,00,f) A0,j, qj, k00,j, j w⇤ w⇤

⌧ 0.06

4kt/kss

33, 59% 21, 18%

11, 38

¯ mt

4wl,t/wl,ss

4wh,t/wl,ss

kt/kss

¯

4mt/mt

4wh,t/wh,ss

wh,ss m¯t m¯t m¯t m¯t

pt= pt> 0 pt= 0

pt+1= pt+1> 0 q 0

pt+1= 0 1 q 1

• •

qpt+1 pt 1 + rt+1 • b0 < u • b0= u • b0> u

•

• •

B 1

U gyt, dot+1 = g y t + dot+1 do t+1= stRt+1 g_ty= ✓wt+ qtbt bt 1 st= (1 ✓) wt t = 0 d0= R0k0 Yt= Z1 ztL1 ↵t Kt↵ 0 < Z < 1 zt✏{0, 1} zt t Z

• yt= Zk↵t • yt = k↵t kt ⇡t= max Lt f (kt, zt) = Z1 ztkt↵ wtLt wt= 8 < : (1 ↵) k↵ t zt= 1 (1 ↵) Zk↵ t zt= 0 ⇡t= 8 < : ↵k↵ t zt= 1 ↵Zk↵ t zt= 0 dt= 8 > > > < > > > : ↵k↵ t ((1 ↵) (1 ✓)) k↵ 1t+1 zt= 1; zt+1= 1 ↵Zk↵t ((1 ↵) (1 ✓)) k ↵ 1 t+1 zt= 1; zt+1= 0 ↵Z2_k↵ t ((1 ↵) (1 ✓)) k↵ 1t+1 zt= 0; zt+1= 0 zt= 0 zt+1= 1

d0= 8 < : ↵k↵ 0 z0= 1 ↵Zk0↵ z0= 0 t t + 1 1 It= st kt+1= st st= (1 ✓) wt kt+1= (1 ✓) (1 ↵) Z1 ztkt↵ k0 r⇤ qt

E [zt] E [zt+1] ...E [z1] = 1 qt= ! = 1 1 + r⇤ E [zt] E [zt+1] ...E [z1] = 0 qt= 0 8 > > > < > > > : qt= ! E [zt] E [zt+1] ...E [z1] = 1 qt= 0 E [zt] E [zt+1] ...E [z1] = 0 qt= 0 zt 1= 0

max bt t=1_X t=t0 t t0 (gty) + d y t+1 gt= ✓Z1 zt(1 ↵) k↵t + E [qt] zt 1bt E [zt] zt 1bt 1 dt+1= (1 ✓) (1 ↵) ↵Z1 ztkt↵kt+1↵ 1

kt+1= (1 ✓) (1 ↵) Z1 zt+1kt↵ kt, bt 1, zt 1 bt t t0_z t 1E [qt] = t+1 t 0 ztE [zt+1] E [qt] = E [zt+1] zt zt 1= 1 kt bt 1 kt+1 qt Vt(zt,{bt}) = t=_X1 t=t0 t t0 _(gy t) + d y t+1 0 < < 1 0 < < 1 t0 Vt ⇣ 1,_{b⇤t} t=1 t=t0 ⌘ Vt(0, 0) {b⇤ t} t=1 t=t0

{bt= bt 18t} ⇢ bt= [(1 ✓) (1 ↵)] ↵ ↵t+1 1 ↵ _k↵t 0 b 18t

↵k↵ 0(1 Z) + t=_X1 t=0 t_b t(qt ) t=_X1 t=0 t ✓✓ (1 ↵) ✓ [(1 ✓) (1 ↵)]↵1 ↵t↵ _k↵t 0 ◆↵◆ ✓ 1 Z11 ↵t↵ ◆◆ t=_X1 t=0 t ✓✓ (1 ↵) ✓ [(1 ✓) (1 ↵)]↵1↵t+2↵ _k↵t 0 ◆↵◆ ✓ 1 Z11↵t+2↵ ◆◆ b 1 {bt= 08t} k0 ✓ (1 ↵) k↵ 0 b 1 k_⇤ k0 k_⇤ t 1 t + 1 ! t 1 t + 1

t=_X1 t=0 t ✓ [(1 ✓) (1 ↵)]↵1 ↵t↵ _k0 ◆↵t = 1 1 k0> k0 kb 1 0 k0dg k_⇤ k₀dg • k0 < kb0 1 < k dg 0 • kb 1 0 < k0< k0dg • kdg0 < k0< kb0 1 • k0 > kdg0 k0 > kb0 1 k0 < k 0 0

• k0 > kdg0 k0 > k0b 1 k0 > k

0

0

kss= [(1 ✓) (1 ↵)]11↵ t0 t0 t=_X1 t=t0 t t0 ✓ (1 ↵) ✓ ✓ [(1 ✓) (1 ↵)]11 ↵t↵ _(kss₎↵ t◆↵◆ ✓ 1 Z11 ↵t↵ ◆ + t=_X1 t=t0 t t0 ✓ (1 ↵) ✓ ✓ [(1 ✓) (1 ↵)]↵1↵t+2↵ _(kss₎↵ t+2◆↵◆ ✓ 1 Z11↵t+2↵ ◆ + = b (kss) ✓_{1 !} 1 ◆ k0 > kss {bt= b (kss)8t 0} 0 < t < t0

t=_X1 t=1 t 1 ✓ (1 ↵) ✓ ✓ [(1 ✓) (1 ↵)]11↵t↵ ⇣ k↵t⌘◆ ↵◆ ✓ 1 Z11 ↵t↵ ◆ + t=_X1 t=1 t 1✓_{(1 ↵) ✓}✓_{[(1 ✓) (1 ↵)]}↵1↵t+2↵ _(k 0)↵ t+2◆↵◆ ✓ 1 Z11↵t+2↵ ◆ + = b k01 ✓ 1 ! 1 ◆ k0< kss bt= b k01 8t 0

zt = zt+1 zss= (1 m) q¯ (1 q) (1 m) n (1 m)¯ kt= kt+1 kt= ⌧ A0z +↵ t ! 1 1 " = ' (zt) kt= 0 B B @ ⌧ A0 ✓ _{+↵ 1} ↵+ ◆ zt 1 C C A ↵+ ↵+ " = ⇠ (zt) zt,f z +↵ " t 1= 1 "_{⌧ A0}

• kf or⇤ = ⌧ A0(zss) +↵ nh(1 m) z¯ ss+ ⇥ ! 1 1 " • k⇤_inf = 0 B B @ ⌧ A0 ✓ +↵ 1 ↵+ ◆ zss nh(1 m) z¯ ss+ ⇥ 1 C C A ↵+ ↵+ " dkt d⌧ = ✓ (1 ⌧ )2(1 ↵) ◆ z ↵+ " f,t ⌧ dkt+1 d⌧ = A0ztk " ↵+ t _⇤ ✓ +↵ 1 ↵+ ✓ 1 +⌧( +↵ 1 ↵+ ) 1 ⌧ ◆◆

+↵ 1 ↵+ ✓ 1 ⌧₍ 1 ↵+ ) 1 ⌧ ◆ t + 1 wh,t+1 " ↵ + ⌧ 1 (1 ↵)" + (↵ + ) (↵ + )2 (1 ⌧ ) 1 > 0 (1 m¯t) = (1 ⌧ ) ↵A0z↵+f,t 1kt" w (1 x) + (1 ⌧ ) ↵A0z↵+_f,t 1kt" zt,f • zt+1=(1 m¯t+1) (nzt+ q) (1 m) (1 q)

• kt+1= ⌧ A0z_f,t+↵k"t (1 m¯t+1) (ztnh+ nlq) + nl((1 m) (1 q)) zt,f • kt+1= 0 @ ⇤ (1 ⌧ ) ↵A0 ⇣ qzt↵+ 2+ zt↵+ 1(n (1 q) (1 m)) ⌘ 1 A 1 " = (zt) • kt+1= 0 B @ ⇤ (1 ⌧ ) ↵A0(B) ↵+ 1 ↵+ ⇣_qz 1 t + (n (1 q) (1 m)) ⌘ 1 C A ↵+ " = (zt) ⇤ = wt(1 xt) (1 q) (1 m) {kt+1, zt} 1 m¯t,f = (1 ⌧ ) ↵A0z↵+t 1k"t wt(1 xt) + (1 ⌧ ) ↵A0z↵+t 1k"t

• nl((1 m) (1 q)) kt+ 1 m¯t,f kt⇤ (nhzt+ nlq) ⌧ A0zt+↵kt"= 0 • 0 B B @ (1 ⌧ ) ↵A0 ✓ _{↵+ 1} ↵+ ◆ kt wt(1 xt) + (1 ⌧ ) ↵A0 ✓ ( ) ↵+ 1 ↵+ ◆ k " ↵+ t 1 C C A (nhzt+ nlq) + k ↵+ " ↵+ t nl((1 m) (1 q)) ! = ⌧ A0 ↵+ 1 ↵+ _zt zt zt+1 {kt+1, zt} kt= ⌧ Yf, 1 ((1 m¯t,i) (nhH 1+ qnlL 1)) + nl(1 m) (1 q) L 1 ¯ mt,i= wt(1 xt) wt(1 xt) + (1 ⌧ ) ↵A0 ↵+ 1 ↵+ _(kt) " ↵+ = f (mt,i) f (mt,i)

• limm¯t!0f ( ¯mt) = wt⇤ w⇤ t+(1 ⌧ )↵A0 ↵+ 1 ↵+ _lim mt,i!0kt ! _" ↵+ =< 1 • limm¯t!1f ( ¯mt) = w(1 x) w⇤_t+(1 ⌧ )↵A0 ↵+ 1 ↵+ _lim mt,i!1kt ! " ↵+ =< limm¯t!0f ( ¯mt) < 1 ¯ mt 1 f (mt,i) f (mt,i) [0 1] df (mt,i) dmt,i = " ↵+ w⇤t(1 ⌧ ) ↵A0 ↵+ 1 ↵+ _(kt)↵+" ✓ w⇤ t + (1 ⌧ ) ↵A0 ↵+ 1 ↵+ _(kt) " ↵+ ◆2⇤ (nhH 1+ qnlL 1) (((1 m¯i,t) (nhH 1+ qnlL 1)) + nl(1 m) (1 q)) [0 1] zt,f zt zt= 1 m¯t,f (nz 1+ q) (1 q) (1 m) ¯ mt,f = w⇤ t w⇤ t + (1 ⌧ ) ↵A0(zt)↵+ 1(kt)" = f mt,f • limmt!0f ( ¯mt) = w⇤t w⇤_t+(1 ⌧ )↵A0 lim mt,f!0zt !↵+ 1 lim mt,f!0kt !" = C < 1 f (mt,f) mt

• ↵ + < 1 lim mt,f!1 1 m¯t,f (nzt 1+ q) (1 q) (1 m) !↵+ 1 e (↵+ 1)ln (1 mt,f_{(1 q)(1})(nzt 1+q_m) ) ! = +1 ↵ + < 1 lim mt,f!1 f ( ¯mt) = w ⇤ t w⇤ t +1 = 0 [0 1] df (mt,i) mt,i = (1 ⌧ ) ↵A0(↵ + 1) ⇣ _nz 1+q (1 q)(1 m) ⌘ _{(1 m} t,f)( nz 1 +q) (1 q)(1 m) !↵+ 2 (kt)" ⇣ w⇤ t + (1 ⌧ ) ↵A0(zt)↵+ 1(kt)" ⌘2 " (1 ⌧ ) ↵A0 ✓ (1 mt,f)nz 1+q (1 q)(1 m) ◆↵+ 1! (kt)"_{(((1 m} (nhHt 1+qnlLt 1) t,f)(nhHt 1+qnlLt 1))+nl(1 m)(1 q)L 1) ⇣ w⇤ t + (1 ⌧ ) ↵A0(zt)↵+ 1(kt)" ⌘2 ↵ + < 1 • ↵ + > 1 e (↵+ 1)ln (1 mt,j_{(1 q)(1})(nzt 1+q_m) ) ! = 0 lim mt,f!1 f ( ¯mt) = w ⇤ t w⇤ t = 1

↵+ > 1 mt= 1 d (zt) d⌧ = 1 " ✓ ⇤ (1 ⌧ ) ↵A0⌅ ◆1 " " _{(⇤↵A0⌅)} ((1 ⌧ ) ↵A0⌅)2 ⌅ =⇣qz↵+_t 2+ z↵+_t 1(n (1 q) (1 m))⌘ ⌅ d (zt) d⌧ = ↵ + " 0 @ ⇤ (1 ⌧ ) ↵A0 ↵+ 1 ↵+ _⇧ 1 A ↵+ " " 1 ↵+ ✓ (1 ⌧ ) 1 ↵ ↵+ ◆ ⌥ ✓ (1 ⌧ ) ↵A0 ↵+ 1 ↵+ _z 2 t ⇧ ◆2 ⇧ = qz3 t + (n (1 q) (1 m)) ⌥ = qz_t 1+ (n (1 q) (1 m))

q, nl, ↵, kss

q nl ↵ kss

kss

q nl ↵ kss

kss = 536, 84

" ⌧ (+0, 01) ¯

¯ mt

4wh,t/wss

4wh,t/wss

wh,ss m¯t m¯t m¯t m¯t

kt+1= [(1 ✓) (1 ↵)] kt↵ k0 k1= [(1 ✓) (1 ↵)] k↵0 k2= [(1 ✓) (1 ↵)] k↵1 = (1 ✓) (1 ↵) [(1 ✓) (1 ↵) k↵ 0] ↵ = [(1 ✓) (1 ↵)]↵↵+1k↵2 0 k3= [(1 ✓) (1 ↵)] k↵2 = (1 ✓) (1 ↵)h[(1 ✓) (1 ↵)]↵↵+1k↵2 0 i↵ [(1 ✓) (1 ↵)]↵2+↵+1k↵2 0 kt= [(1 ✓) (1 ↵)] 1 ↵t 1 ↵ _k↵t 0 t = 1 k1= [(1 ✓) (1 ↵)] k0↵= [(1 ✓) (1 ↵)] 1 ↵ 1 ↵_k↵t 0

kt= [(1 ✓) (1 ↵)] 1 ↵t 1 ↵ _k↵t 0 kt+1= [(1 ✓) (1 ↵)] 1 ↵t 1 ↵ _k↵t t kt+1= [(1 ✓) (1 ↵)] ✓ [(1 ✓) (1 ↵)]11↵t↵ _k↵t 0 ◆↵ kt+1= [(1 ✓) (1 ↵)] 1 ↵t+1 1 ↵ _k↵t+1 t