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论文研究-老龄化下人口政策与经济增长关系研究.pdf

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    38 2 2018 2  Systems Engineering — Theory & Practice Vol.38, No.2 Feb., 2018 doi: 10.12011/1000-6788(2018)02-0337-14 : F061.2 : A  1,  2,  2 (1.  , 100084; 2. Æ, 100871)  , !"$%&, ’() **-$%. 023*4, 56  ; >?@, "B&#- CD%BEF . &I ’% J()K0MNO*+,&, M*&PQ, /R4’( )KT#234, V)5-$%68; W’9Y Z[\8P. %_‘B$% %C 3EaFFbJ, c3HddIe\<*, Jg’  , %$%4Li’j.  ; Mk); ; $%% Study on the relationship between population policy and economic growth in the context of aging WEI Jiang1, NI Xuanming2, HE Aichen2 (1. School of Economics and Management, Tsinghua University, Beijing 100084, China; 2. School of Software and Microelectronics, Peking University, Beijing 100871, China) Abstract This paper constructs population-economy model under the background of population aging degree deepening to explore the effects of aging, two-child policy and postponed retirement on economy in the long term. Starting from the micro decision making of individuals and companies, capital stock in steady state conforms to the golden rule in general equilibrium. The model in this paper is the improvement and innovation of neoclassical growth theory. By introducing newborn growth rate and life expectancy into the model and using numerical simulation method, we find that birth control and longevity are both the causes of ageing but have different impacts on economy. In the other hand, encouraging birth and postponed retirement can both adjust the age and employment structure but have opposite effects on economy. The long-term economic growth depends on exogenous technological progress. To solve the problem of aging, government should not only rely on population and employment policy, but also focus on improving the ability of innovation. Keywords population aging; two-child policy; postponed retirement; economic growth 1  , 65  7%  . 2001 ,  2001 2005 ,  0.59%; 2006 2010 ,  0.94%; 2011 2015 ,  1.3%. ,  : 2017-05-24 lmÆ: (1991–), ,  , , : , E-mail: weij.09@sem.tsinghua.edu.cn; :  (1984–), , , , , : , , E-mail: nixm@ss.pku.edu.cn;  (1996–), , Æ , , : , E-mail: heaichen@pku.edu.cn. n: , , .  [J]. , 2018, 38(2): 337–350. : Wei J, Ni X M, He A C. Study on the relationship between population policy and economic growth in the context of aging[J]. Systems Engineering — Theory & Practice, 2018, 38(2): 337–350.
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    2 , :  339 *!, $ “BJ!” /)’!. !G Solow !, 7I40 , JD1#B-. Cass[3] Koopmans[4] Ramsey[5] , ("#"KÆ,, ’1($K*1, Solow !#(-0, !G RCK ! Ramsey !. ("#"KÆ,K, Samuelson[6] 8 % !K9Æ,,  ! (saddle) 10<=, !)’L 4M)’. Ramsey !L8 Solow !I4, -JD1#B -. 0-B-0, M%">. Arrow[7] C  (learning by doing) !, 0 B  M, 8*!8 Æ,, B% AK !8. Uzawa[8] )*M! ?N!, !  M. Romer[9,10] Lucas[11,12] ??/ !!-, $0 I #, ),N%JB, K$0 ! ,  , ?.9 Arrow-Romer ! Uzawa-Lucas !. 2.2 !"# !r1 Ramsey !Æ’"#"’1$", C’1$8, (infinite horizon), B(% 9316, 932P%Æ,. Samuelson[13] Diamond[14] 8"#"’1$8, Æ,, (, 0 "#"’1$3, ?$, ?#Q , (K’1, ?=1)C)* OLG !. 0# %: O5. %?$ OLG ’(%, L. 39?/8L, &/I , ÆP9QQ/%’1<=. >P, M% , B,%%R:0QQ, Æ  !(R%, ,%  OLG &-, Æ, , )-. #, OLG !, *". < [15]  0 $0, S-S, <-5TB, $05T , 16(T6:+M. =Q [16] C 8%U7 10<=, S-!R, "#"’1$LVB, ))*’ 1(1B.  [17] 0!(, "=0, %, S- "WNVR, "WN";(", 5T$J %/%. ? [18] %%"!"$. %, TX $&,8"%, 3933/48, #:$ , )8. XY [19] =) 75 $!"/$!, %1 Z . :D4(EL,  1∼2 9V1-[W. ) "(T: 1,, SW:#; Z ,,   X Z B, 725Z . A9[20] =)/\0 -!"/$ !, % -, 5,. *!5, C: -, -W. 05,  C: -(Y7! B, # . 3 %&’) 3.1 ;+- RCK K% , @,K6, -,/0, %3, 2K"#" K, P/K]’8; ?$ OLG !WZI, %,  ?, /% , /%’<, $, Q, Q , C!"
    340       38 ’1, % $C!", C!! , "4/$. 2% Futagami Nakajima[21] @?:,/$098>*/%’1(, ’, /%0, m, % t Q,  0 m /%, 0 #Q/%^ , m #Q/% 9<. &!, %R N, 5,  0 m 8C, [-\ N/m. /%, d, /% 0 d 8C!", 6"S, %’1(, d m 8, ’10X(! N7 . $ ÆC , %"3 L = dN/m, 3 O = (m − d)N/m. S-8 E, %, /% /%<, <, ]’=*)’. /! Æ5 (1) @Q*+: (1) B(x, t) #QQ x /% t Q/!, C(x, t) #QQ x /% t Q’13, r(t) #Q t Q -, W (x, t) #Q t Q4!. /%#F, 4!J: ˙B(x, t) = W (x, t) + r(t)B(x, t) − C(x, t) W (x, t) = W (t), x ≤ t ≤ x + d 0, x + d < t ≤ x + m /% -,/!, P,G>, J5 (3): B(x, x) = B(x, x + m) = 0 ’P[,"/! b(x, t) = B(x, t)/A(t), /!*+ I V.5: ˙b(x, t) + b(x, t)g = ω(x, t) + r(t)b(x, t) − c(x, t) , P[,"/! Æ5 (5)∼(7) ?:*+: ˙b(x, t) = ω(x, t) + (r(t) − g)b(x, t) − c(x, t) ω(x, t) = ω(t), x ≤ t ≤ x + d 0, x + d < t ≤ x + m b(x, x) = b(x, x + m) = 0 /%\%82% CRRA !, -%.5: u(C(x, t)) = C(x, t)(1−θ) 1 − θ , θ > 0 θ #Q_]6, θ JK, ’1W@(D4%BJ], /%JLR^’1>$8 . Q x /% %:  C(x, t) = A(t)c(x, t), A(t) = A(0)egt : x+m e x −ρ(t−x) C(x, t)(1−θ) 1 − θ Q x /%#F, A(0)1−θeρx R, */: max x+m e x −ρ(t−x) C(x, t)(1−θ) dt 1 − θ x+m e x dt = A(0)1−θeρx −(ρ−g(1−θ))t c(x, t)(1−θ) 1 − θ dt (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎩ s.t. x+m −(ρ−g(1−θ))t c(x, t)(1−θ) x e max ˙b(x, t) = ω(x, t) + (r(t) − g)b(x, t) − c(x, t) ω(x, t) = 1 − θ ω(t), x ≤ t ≤ x + d 0, x + d < t ≤ x + m dt b(x, x) = b(x, x + m) = 0 ’ Hamilton 8 H = e−(ρ−g(1−θ))t c(x,t)(1−θ) 1−θ + λ[ω(x, t) + (r(t) − g)b(x, t) − c(x, t)], : = r(t) − ρ − θg ˙c(x, t) c(x, t) θ $ : c(x, t) = c(x, x)eP (x,t)
    Æ, f 7<, lim k→0 (k) > 0, f (k) = ∞, f (k) = 0. lim k→∞ f -! D4 , : P[,"! #Q: 3.3 / 0C % %’1!, : r(t) = f (k(t)) ω(t) = f(k(t)) − k(t)f (k(t)) (19) (20) 341 (14) (15) (16) (17) (22) (23) ∗ (24) (25) , :  2 7, P (x, v) = θ v x r(t)−ρ−θg dt. #, ’ R(x, v) = x (r(t) − g)dt. 5 (5) ?DW e−R(x,t) Q x B x + m : −R(x,v)dv − $ b(x, x) = b(x, x + m) = 0, c(x, t) = c(x, x)eP (x,t), : −R(x,v)|x+m b(x, v)e ω(v)e x+d = x x x v x+m c(x, v)e −R(x,v)dv 0 = ω(v)e x+d x −R(x,v)dv − x+d x x+m x c(x, x) = x+m c(x, x)eP (x,v)−R(x,v)dv x ω(v)e−R(x,v)dv eP (x,v)−R(x,v)dv c(x, t) = x+d x x+m x ω(v)e−R(x,v)dv eP (x,v)−R(x,v)dv eP (x,t) #5?:/% $0[QP[,/’1, 5 ω(v)e−R(x,v)dv 4"#/ % 3U, /%7%, 931a’1, 2@$Æ4_. 3.2 A.- x+d x 8/: Y (t)! K(t)" L(t) B A(t), 83S8 t 8, 7-%.5: (18) (k) < 0,  , f(0) = 0. #, V.58 f J Inada Y (t) = F (K(t), A(t)L(t)) (21) 7, ˜c(t) #Q t Q%@,/%P[,/’1!, A(t) ˜c(t) #Q t Q%@, /%4’1!. I V.5, P[,"! 3+: ˙K(t) = F (K(t), A(t)L(t)) − A(t)˜c(t) ˙k(t) = f(k) − ˜c(t) L(t) − k(t)g )!, ˙k(t) 4!LA1!, 4! f(k) − ˜c(t) L(t), Æ,, DZ, 1 ! k(t)g. $\R N/m,  x Q @,/% t QP[,/’1! N/m · c(x, t). t Q, %/% Q t − m, /% Q t, t Q %@,/%P[,/’1!: N m c(x, x)eP (x,t)dx c(x, t)dx = ˜c(t) = N m t t−m t t−m )* ˙k = 0, k R, #, - r P[,"! ω R, 8 t , r ∗, 5 (23) I: ω ˜c(t) = N m · ω ∗ ∗( r θ − ρ (e( r∗ θ θ − r − ρ ∗)d − 1)(e( r ∗−ρ−θg ∗)(e(g−r −r∗)m − 1)(g − r∗)( r∗−ρ−θg )m − 1) ) θ θ θ $ L = dN m , : ˜c(t) L(t) ∗ = ω d ∗ · ( r θ − ρ (e( r∗ θ θ − r − ρ θ ∗)d − 1)(e( r ∗)(e(g−r −r∗)m − 1)(g − r∗)( r∗−ρ−θg ∗−ρ−θg θ ) )m − 1) θ
    342       38 )* ˙k = 0, "5 (22) : − ρ (26) (e( r∗ ∗), 72@ {ρ, θ, d, m, g} #"’. ]’ ∗)(e(g−r −r∗)m − 1)(g − r∗)( r∗−ρ−θg ) − ω d ∗) − k f(k ∗ = f(k ∗)d − 1)(e( r )m − 1) − r − ρ ∗ = f ∗), ω ∗−ρ−θg · ( r (k (k = k ∗ ∗ ∗ ∗ g ) ∗ θ θ θ θ θ θ 5, r C., (=@, 0. 3.4 !1EF34 f =/8D1 -<-8. ?:&!, B(% Æ C, -<-*]’, #%*]’=*)’,    . )K 1949–1957 8,  ’ (1C :O‘7<, <-[B, -4 :(1, # /-: :>*. 1949 , 5.42 , 1957 ,  6.47 , 8 8\! 1.05 , 9 )K>C:b. ), =8$ “ 7” “<7” c1’. J(0) %b 3QQ “ 7” , %$0QQ “<7” , ?$ A%%=. -%, 62%Æ,, ’b 3 ˙J(t) J(t) = n 1’. n #Q-, B( n > 0, a-b 3!; B J, 78$5 ( n < 0, a-b 3L. cZC, !R 3 6 “7” , ! n < 0, 4, ! @, n > 0 P %. Æ,Q 3 J(0), # t Q 3 #Q J(t) = J(0)ent. t  Q, % 0 m ,  Q t t−m, Q x \ J(0)enx, , ˙N (t) % N(t) = N (t) = (nent−nen(t−m)) (ent−en(t−m)) = n. t Q, %" 0 d ,  Q t t−d, Q n (ent − en(t−d)). x \ J(0)enx, , %"3 L(t) = (nent−nen(t−d)) (ent−en(t−d)) = n. , % %"-#Q (nen(t−d)−nen(t−m)) n (en(t−d) − en(t−m)) 3-#Q O(t) = (en(t−d)−en(t−m)) = n. (ent−en(t−d)) (ent−en(t−m)) = 1−e−nd $ , %"] L(t) N (t) = l(t) = N (t) = 1 − o(t), l(t) #Q"], o(t) #Q]-, l(t) o(t) 8 t , I l o. %-$0Bd=, ∂l ∂n = de−nd(1−e−nm)−me−nm(1−e−nd ) > 0, $ ^, "’$0, B(3L]’8, # n (ent − en(t−m)). %-#Q t−d t−m J(0)enxdx = J(0) t−d J(0)enxdx = J(0) t−m J(0)enxdx = J(0) 1−e−nm . ’ L(t) ˙L(t) L(t) = ˙O(t) O(t) = n J(0) n J(0) n J(0) n J(0) J(0) J(0) J(0) n n n n t t (1−e−nm)2 =; 3,3!L, =. ∂l 0, $ ^, $0J, "J, JR. 3.5 !56H/ 0C ∂m = n(e−nd−1)e−nm (1−e−nm)2 < ]’C7<&!%L, /\@7<] 0, #)*6I!.  x Q /%#F, 7 ’1*de/ %5 (16) 5 (17) ?:. x Q /%3 J(x) = J(0)enx,  x Q @,/% t QP[,/ ’1! J(0)enxc(x, t). t Q, %/% Q t − m, /% Q t, t Q%@,/%P[,/’1!: t t−m t t−m ˜c(t) = J(x)c(x, t)dx = J(0)enxc(x, t)dx % %’1!, : ˙K(t) = F (K(t), A(t)L(t)) − A(t)˜c(t) , P[,"! 3+: ˙k(t) = f(k) − ˜c(t) L(t) − k(t)(g + n) (27) (28) (29)
    2 , :  )!, ˙k(t) 4!LA1!, 4! f(k) − ˜c(t) L(t), Æ,, DZ, 1 ! k(t)(g + n). )* ˙k = 0, k R, #, - r P[,"! ω R, 8 t , r ω 343 ∗ θ ∗ ω c(x, x) = ∗)(e(g−r ∗, c(x, x)c(x, t)˜c(t) I: ∗( r (e( r∗ ∗)d − 1) − r −r∗)m − 1)(g − r∗) ∗−ρ−θg − r ∗)(e(g−r θ −r∗)m − 1)(g − r∗)( r∗−ρ−θg − ρ − r ∗)d − 1) −r∗)m − 1)(g − r∗) · e( r − ρ θ − ρ ∗)(e(g−r ∗)d − 1)(e( r − ρ (e( r∗ ∗( r (e( r∗ − ρ θ − ρ c(x, t) = )(t−x) ∗−ρ−θg ∗( r ω ∗ ∗ θ θ θ θ θ θ θ θ θ θ θ (30) (31) (32) −n)m − 1) − n) ˜c(t) = J(0)enxc(x, t)dx = J(0)ent · ω t t−m n (ent − en(t−d)), : , L(t) = J(0) ∗)d−1)(e( r ∗ − ρ −r ∗( r (e( r∗ θ − ρ θ θ n (ent − en(t−d)) J(0) ∗−ρ−θg ∗)(e(g−r −r∗)m−1)(g−r∗)( r∗−ρ−θg J(0)ent · ω = θ θ θ ˜c(t) L(t) −n)m−1) −n) ∗( r ω (e( r∗ ∗ − r ∗)(e(g−r − ρ ∗)d − 1)(e( r θ −r∗)m − 1)(g − r∗)( r∗−ρ−θg − ρ θ θ θ ∗−ρ−θg θ −n)m − 1)n − n)(1 − e−nd) = θ (33) , $P[,"! 3+ : θ θ θ θ ∗ f ∗ ∗ ∗−ρ−θg (k (k − r ∗ = f ∗( r ) − ω (e( r∗ ∗ = f(k −n)m − 1)n − n)(1 − e−nd) ∗)(e(g−r − ρ ∗)d − 1)(e( r θ −r∗)m − 1)(g − r∗)( r∗−ρ−θg − ρ ∗) − k ∗ f(k ∗), ω (34) ∗), 72@ {ρ, θ, d, m, n, g} #"’. 5, J 5, r ∗) = g + n, (" (, , 7c (^:)*., C% (k f (, Æ,8_ - e.F!, (-%^). "’@‘, % (+Æ P[,"! 35 (34),  k1 k2 ?/0. B 1 @Q, g k1 )’, k2 )’, -%, ! +5 (29), k k2 -R"KKE, ˙k(t) < 0, # k  k2; k k2 -"KKE, ˙k(t) > 0, # k  k2. , k2 )’, , k1 )’. %10<=P[,"! 3 k2. (g + n) = k θ k1 k2 =0.6 n=－0.0123 g=0.05 k =0.01 =0.75 d=37 m=48 =0.01 =0.75 d=37 m=56 =0.01 =0.75 d=37 m=70 7 1 K89:;sM< 7 2 K89:;sM
    344       38 Q x /% %: x+m −ρ(t−x) C(x, t)(1−θ) x e U(x) = (, > A(0)(1−θ)eg(1−θ)x A + ˙L 1−θ K = ˙A ˙Y Y = ˙K 1 − θ  ! - B-. dt = A(0)(1−θ)eg(1−θ)x 1 − θ ∗ · c(x, x)(1−θ) · (e( r θ ( r∗ θ − ρ −r − ρ ∗)m − 1) − r∗) (35) θ θ 3/% Q7c#@,, , 1. L + ˙k k = g + n, ^, $D1- ! : K N = ALk N = Ak · 1 − e−nd 1 − e−nm , ,! ,"!  ˆk = k · L (-: = 1 − = 1 − ω A(t)˜c(t) ∗ s θ = Af(k) · 1 − e−nd 1 − e−nm Y N = ALf(k) N N = k · 1−e−nd 1−e−nm . (36) A(t)L(t)f(k∗) ∗ = g + n ",  s ∗ ∗( r ∗)(e(g−r ∗)d − 1)(e( r ∗−ρ−θg − r − ρ θ −r∗)m − 1)(g − r∗)( r∗−ρ−θg f(k∗)(e( r∗ − ρ ∗ ∗) (k ∗ = k f . #@8!083 f (k∗) −n)m − 1)n − n)(1 − e−nd) (37) θ θ θ θ , r Rg, %(+Æ" . 4 u)? 4.1 F@Q 4.1.1 ARSB CU (, 93" 8-%.5, 6Æ,8_ - e.F!,  !8J87<, -%.5 F (K, AL) = K α(AL)1−α, 0 < α < 1, % V.5#Q f(k) = kα, 0 < α < 1. % , k8_ - e.F!, ]Æ,! ge- 0.3∼0.8 )8. Wf‘a [22] 2%! ge- 0.489, Vh [23] M! ge- 0.609. , 6Æ’! ge- α = 0.6. ,%B-X, l. Vh [23] X 1978–1998 B- 2.8%, bWX [24] X 1979–2007 B- 3.588%. , 6Æ’B- g = 5%. Cooper Kaplanis[25] Ferson[26] %C, _]6 K 0.5, 10, , 6Æ, θ = 0.75, , Æ,’1$8U7 ρ = 0.01. 4.1.2 !w ’b 3 J(t), n #Q-, B( n > 0, a-b 3!; B( n < 0, a-b 3L. 1987 )Kb3B, a 10 )’, 6% 1987–2015 1-"# n,   n = −0.0123. 2015 "d“ 1nS"?1=9” Si, 16∼19 11 1.8%, 20∼24 11 8.7%, 25∼29 1 1 12.5%, 55∼59 11 7.1%, 60∼64 11 5.5%, 65 11 4.4%, # “ 1"?eX1=9” Si, 60∼64 11 2.7%, 65 11 1.7%. , 6D 20 , 57 . #, M, 2015 1$0 76.34 . $ WZ/% "WN), , kK, , @ d = 57 − 20 = 37, m = 76 − 20 = 56. :, @DB# 1. \fjÆ hfÆ lih jk[] \Æ l] m Y 1 xFGZI α 0.6 g 0.05 θ 0.75 ρ 0.01 n −0.0123 d 37 m 56
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