Vin tage h uman apital, demographi trends and endogenous gro - - PDF document

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vin tage h uman apital demographi trends and endogenous
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Vin tage h uman apital, demographi trends and endogenous gro - - PDF document

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Vin tage h uman apital, demographi trends and endogenous gro wth Raouf Bouekkine Da vid de la Croix Omar Liandro April 20, 2001 {1{ Figure 1: The deline in mortalit y { F rane { Surviv al la ws Soure:

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SLIDE 1 Vin tage h uman apital, demographi trends and endogenous gro wth Raouf Bou ekkine Da vid de la Croix Omar Li andro April 20, 2001
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SLIDE 2 {1{ Figure 1: The de line in mortalit y { F ran e { Surviv al la ws Sour e: Challier and Mi hel (1996). T able 1: Long-run data date Gro wth rate
  • f
p
  • pulation
Y ears
  • f
s ho
  • ling
W estern Europ e U.S. U.S. F ran e 1820-1870 0.8 2.9 2.8 n. a. 1870-1913 0.9 2.1 5.9 5.0 1913-1950 0.6 1.2 9.6 8.3 1950-1973 0.8 1.4 12.9 10.6 1973-1992 0.3 1.0 16.3 13.8 Sour e: Maddison (1995)
slide-3
SLIDE 3 {2{ The set
  • f
individuals b
  • rn
in t:
  • e
nt surviv al probabilit y: m(z
  • t)
= e
  • (z
t)
  • 1
  • >
1;
  • <
0. Upp er b
  • und
  • n
longevit y: m(A) = 0: A =
  • log
( )
  • :
life exp e tan y:
  • =
1
  • +
  • log
( ) (1
  • )
Figure 2: Surviv al la ws 6
  • m(t)
t 6 6
  • m(t)
m(t) t t
  • >
0;
  • =
  • >
0;
  • <
1
  • <
0;
  • >
1 1 1 1 A A
slide-4
SLIDE 4 {3{ Figure 3: Changes in the surviv al la ws 6
  • a
A m(a)
  • +
  • +
  • 1
The size
  • f
the p
  • pulation
at time t: Z t tA
  • e
nz m(t
  • z
)dz =
  • e
n t
  • with
  • =
n(1
  • )
  • (1
  • n=
) n(1
  • )(n
+
  • )
: fertilit y rate: 1=
slide-5
SLIDE 5 {4{ An individual b
  • rn
at time t max Z t+A t (t; z ) m(z t)dz
  • H
(t)
  • Z
t+P(t) t (z t) m(z t)dz ; sub je t to Z t+A t (t; z )R (t; z )dz = Z t+P(t) t+T(t) ! (t; z )R (t; z )dz : Sp
  • t
w ages : ! (t; z ) = h(t)w (z ); Individual's h uman apital: h(t) =
  • H
(t)T(t):
slide-6
SLIDE 6 {5{ A t equilibrium w e ma y rewrite
  • n
tingen t pri es as R (t; z ) = m(z
  • t):
S ho
  • ling
time P(t) = min [T(t)
  • w
(t + P(t)); A℄ :
slide-7
SLIDE 7 {6{ Pro du tion fun tion: Y (t) = H (t): Lab
  • r
mark et equilibrium: w (t) = 1
slide-8
SLIDE 8 {7{ Equilibrium s ho
  • ling
and retiremen t de isions w e dene
  • =
  • Prop
  • sition
1 (i) Ther e exists a unique interior T ? , and P ? =
  • T
? if and
  • nly
if 2 <
  • <
  • ?
. (ii) If
  • >
  • ?
, T ? = T max ( ? ) and P ? = A. (iii) If 1 <
  • 6
2, T ? = P ? = 0. Figure 4: Optimal s ho
  • ling
and retiremen t as a fun tion
  • f
  • T;
P; A

2 η∗ η 20 40 60 80 100

slide-9
SLIDE 9 {8{ Life exp e tan y and
  • ptimal
s ho
  • ling
Prop
  • sition
2 A rise in life exp e tan y in r e ases the
  • ptimal
length
  • f
s ho
  • ling.
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SLIDE 10 {9{ The balan ed gro wth path Pro du tiv e aggregate h uman apital sto k: H (t) = Z tT (t) tP(t)
  • e
n z m(t
  • z
)h(z )dz ; The a v erage h uman apital:
  • H
(t) = H (t)
  • e
n t
  • :
The dynami s
  • f
h uman apital: H (t) = Z tT tP m(t
  • z
) TH (z )
  • dz
slide-11
SLIDE 11 {10{ Life exp e tan y and gro wth Figure 5: Gro wth and life exp e tan y at giv en n (left panel) and at giv en
  • (righ
t panel)

60 80 100 120 140 Λ 0.012 0.014 0.016 0.018 0.02 γ 60 80 100 120 140 Λ 0.006 0.008 0.012 0.014 0.016 0.018 0.02 γ−n

Prop
  • sition
3 A rise in life exp e tan y thr
  • ugh
  • at
given p
  • pulation
gr
  • wth
has a p
  • sitive
ee t
  • n
e
  • nomi
gr
  • wth
for low levels
  • f
life exp e tan y and a ne gative ee t
  • n
e
  • nomi
gr
  • wth
for high levels
  • f
life exp e tan y.
slide-12
SLIDE 12 {11{ P
  • pulation
gro wth and e onomi gro wth Figure 6: P
  • pulation
and gro wth

n 0.005 0.01 0.015 0.02 growth per capita 0.04 0.02 0.02 0.04 n 0.01 0.02 0.03 0.04 0.05 fertility rate

Prop
  • sition
4 Assume that < T < P 6 A. Ther e exists a p
  • pulation
gr
  • wth
r ate nite value n ? su h that the long run p er apita gr
  • wth
r ate
  • f
the e
  • n-
  • my
r e a hes its (interior) maximum at n ? .
slide-13
SLIDE 13 {12{ F rom Malth us to Solo w regime
  • A
T
  • n=0
T= Solo w 5.44
  • .0147
.2531 8.324 73 115 27 .0200 37% Malth us 2.69
  • .0147
.2531 8.324 39 67 13 .0004 33 % Figure 7: F rom Malth us to Solo w

n 0.005 0.01 0.015 0.02 growth per capita

slide-14
SLIDE 14 {13{ The dynami s
  • f
h uman apital detrended h uman apital as ^ H (t) = H (t)e
  • t
; ^ H 00 (t) =
  • (
+
  • )
^ H (t)
  • (
+ 2 ) ^ H (t) + T (1
  • )
h
  • e
( + )T
  • (
+
  • )e
  • T
  • ^
H (t
  • T)
  • e
( + )P
  • (
+
  • )e
  • P
  • ^
H (t
  • P)
i + T (1
  • )
h e ( + )T
  • e
  • T
  • ^
H (t
  • T)
  • e
( + )P
  • e
  • P
  • ^
H (t
  • P)
i : Figure 8: Eigen v alues
  • f
the alibrations \Solo w" and \Malth us" Solo w

0.1 0.08 0.06 0.04 0.02 Real part 2 1 1 2 Imaginary part

Malth us

0.14 0.12 0.1 0.08 0.06 0.04 0.02 Real part 2 1 1 2 Imaginary part

slide-15
SLIDE 15 {14{ Dynami sim ulation Figure 9: Dynami sim ulation
  • f
a drop in fertilit y in Solo w
  • n
  • t