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

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

{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 of p opulation Y ears of s ho oling 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)

{2{ The set of individuals b orn in t : nt � e surviv al probabilit y: � � ( z � t ) e � � m ( z � t ) = 1 � � � > 1 ; � < 0. Upp er b ound on longevit y: m (A) = 0: � log ( � ) A = : � life exp e tan y: 1 � log ( � ) � = + � (1 � � ) � Figure 2: Surviv al la ws m ( t ) m ( t ) m ( t ) 6 6 6 1 1 1 � > 0 ; � < 1 � > 0 ; � = 0 � < 0 ; � > 1 - - - t t t A A

{3{ Figure 3: Changes in the surviv al la ws m ( a ) 6 1 + � � + � � - a A The size of the p opulation at time t : Z t nz n t � e m ( t � z )d z = � e � t � A with n=� n (1 � � ) � � � (1 � � ) � = : n (1 � � )( n + � ) fertilit y rate: 1 =�

{4{ An individual b orn at time t max Z Z t +A t +P( t ) � H ( t ) ( t; z ) m ( z � t )d z � ( z � t ) m ( z � t )d z ; � t t sub je t to Z Z t +A t +P( t ) ( t; z ) R ( t; z )d z = ! ( t; z ) R ( t; z )d z : t t +T( t ) Sp ot w ages : ! ( t; z ) = h ( t ) w ( z ) ; Individual's h uman apital: � h ( t ) = � H ( t )T( t ) :

{5{ A t equilibrium w e ma y rewrite on tingen t pri es as R ( t; z ) = m ( z � t ) : S ho oling time P( t ) = min [T( t ) � � w ( t + P( t )) ; A℄ :

{6{ Pro du tion fun tion: Y ( t ) = H ( t ) : Lab or mark et equilibrium: w ( t ) = 1

{7{ Equilibrium s ho oling and retiremen t de isions w e de�ne � = �� Prop osition 1 (i) Ther e exists a unique interior ? ? ? ? T , and P = � T if and only if 2 < � < � . ? ? ? ? (ii) If � > � , T = T ( � ) and P = A . max ? ? (iii) If 1 < � 6 2 , T = P = 0 . Figure 4: Optimal s ho oling and retiremen t as a fun tion of � T ; P ; A 100 80 60 40 20 η 2 η∗

{8{ Life exp e tan y and optimal s ho oling Prop osition 2 A rise in life exp e tan y in r e ases the optimal length of s ho oling.

{9{ The balan ed gro wth path Pro du tiv e aggregate h uman apital sto k: Z t � T ( t ) n z H ( t ) = � e m ( t � z ) h ( z )d z ; t � P( t ) The a v erage h uman apital: H ( t ) � : H ( t ) = n t � e � The dynami s of h uman apital: Z t � T � T H ( z ) H ( t ) = m ( t � z ) d z � t � P

{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) γ γ− n 0.02 0.02 0.018 0.018 0.016 0.016 0.014 0.014 0.012 Λ 0.012 60 80 100 120 140 0.008 Prop osition 3 A rise in life exp e tan y thr ough � Λ 0.006 60 80 100 120 140 at given p opulation gr owth has a p ositive e�e t on e onomi gr owth for low levels of life exp e tan y and a ne gative e�e t on e onomi gr owth for high levels of life exp e tan y.

{11{ P opulation gro wth and e onomi gro wth Figure 6: P opulation and gro wth growth per capita 0.02 0.015 0.01 0.005 n fertility rate 0.05 0.04 0.03 0.02 0.01 n � 0.04 � 0.02 0.02 0.04 Prop osition 4 Assume that 0 < T < P 6 A . Ther e ? exists a p opulation gr owth r ate �nite value n su h that the long run p er apita gr owth r ate of the e on- ? omy r e a hes its (interior) maximum at n .

{12{ F rom Malth us to Solo w regime � � � � � A T � T = � n =0 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 growth per capita 0.02 0.015 0.01 0.005 n

{13{ The dynami s of h uman apital detrended h uman apital as � � t ^ H ( t ) = H ( t ) e ; 00 0 ^ ^ ^ H ( t ) = � � ( � + � ) H ( t ) � ( � + 2 � ) H ( t ) h� � � T � ( � + � )T � � T ^ + � e � � ( � + � ) e H ( t � T) (1 � � ) � � � i � ( � + � )P � � P ^ � � e � � ( � + � ) e H ( t � P) h� � � � i � T � ( � + � )T � � T 0 � ( � + � )P � � P 0 ^ ^ + e � � e H ( t � T) � e � � e H ( t � P) : (1 � � ) � Figure 8: Eigen v alues of the alibrations \Solo w" and \Malth us" Malth us Solo w Imaginary part Imaginary part 2 2 1 1 Real part Real part � 0.1 � 0.08 � 0.06 � 0.04 � 0.02 � 0.14 � 0.12 � 0.1 � 0.08 � 0.06 � 0.04 � 0.02 � 1 � 1 � 2 � 2

{14{ Dynami sim ulation Figure 9: Dynami sim ulation of a drop in fertilit y in Solo w � � n ����� ����� ����� ����� t ��� �� ��� ���