曲双石：人口老龄化对现收现付养老保险制度的冲击基于三时期OLG模型 的三种解决方案的分析

三.增加养老保险税与个人福利 增加养老保险税与个人福利 水平、初始资本、储蓄 水平、初始资本、
θ =1 ↓
rt = α kt α −1
α (1 + n )( ρ 2 + 3 ρ + 3) r = 1−α
*
最优均衡实际利率：
假定一个封闭的经济，不存在技术进步和折旧 ↓
yt = f (kt ) = ct + st = ct + ∆k + nkt
经济处于稳态时，资本存量不发生变化
↓ ∆k = 0 ct = f (kt ) − nkt ↓
1
工作期的储蓄可以全部转化为退休期的资本→
K t + 2 = Tt +1 Lt +1
→
K t + 2 K t + 2 Lt + 2 Tt +1 = = ⋅ = kt + 2 (1 + n) Lt +1 Lt + 2 Lt +1
1 1 + rt + 2 θ 1 [ ] ⋅ (1 + ρ ) 2 1 + rt + 2 (1 − α )kt +1α kt + 2 (1 + n) = 1 1 1 + rt + 2 θ 1 θ 1 1+ ( ) +[ ] ⋅ 2 1+ ρ (1 + ρ ) 1 + rt + 2
→ r = f ′(k ) = α k α −1 t t t ↓
1 + rt + 2 θ 1 [ ] ⋅ 2 (1 + ρ ) 1 + rt + 2 Tt +1 = (1 − α )kt +1α 1 1 1 + rt + 2 θ 1 θ 1 1+ ( ) +[ ] ⋅ 2 1+ ρ (1 + ρ ) 1 + rt + 2
→
↓在稳态均衡时
k 1 = k 2 = ... = k t + 1 = k *
1 1 + rt + 2 θ 1 [ ] ⋅ 1 2 (1 + ρ ) 1 + rt + 2 1 − α 1−α ) k* = ( 1 1 1 + rt + 2 θ 1 θ 1 1+ n 1+ ( ) +[ ] ⋅ 2 1+ ρ (1 + ρ ) 1 + rt + 2
1 wt +1 1+ n θ c3t + 2 = [ ] 1 1 2 1 θ 1 + n θ 1 (1 + ρ ) 1+ ( ) +[ ] ⋅ 2 1+ ρ (1 + ρ ) 1+ n wt +1 = 1 1 (1 + ρ ) 2 θ 1 + ρ θ 1 [ ] +( ) + 1+ n 1+ n 1+ n
wt +1 wt +1 c1t = = 1 1 1−θ 1 θ 1+ n θ 1 1 1+ ( ) +[ ] ⋅ 1 θ (1 + n) θ ) + 1+ ρ (1 + ρ ) 2 1 + n 1 + ( 2 1+ ρ (1 + ρ )θ
1 wt +1 wt +1 1 θ c2t +1 = ( ) = 1 1 1−θ 1 θ 1+ n θ 1 1+ ρ 1 1+ ( ) +[ ] ⋅ 1 −θ (1 + n) θ 1+ ( ) + 1+ ρ (1 + ρ ) 2 1 + n 1 1+ ρ (1 + ρ )θ
↓ 个人效用最大化同样要求实际利率等于人 口增长率
由
α (1 + n)( ρ 2 + 3ρ + 3) r* = 1−α
ρ 可知当 α 、或 n 足够小 ，即资本在产出中的 比重足够小，或者个人非常在乎退休期的 α ( ρ + 3ρ + 3) n< 消费,或者人口增长率足够小 （ 1 − α ( ρ + 3ρ + 4) ） 时，可能存在
c1t = K t
c2t +1 = wt +1 − Tt +1 − St +1 − K t
c3t + 2 = Tt + 2 + (1 + rt + 2 ) St +1
↓
Tt + 2 + wt +1 − Tt +1 1 + rt + 2 K t = c1t = 1 1 1 + rt + 2 θ 1 θ 1 1+ ( ) +[ ] ⋅ 2 1+ ρ (1 + ρ ) 1 + rt + 2
Tt +1 ：个人缴纳的完全基金型养老金
rt + 2 ：退休期的利率
↓
c3t + 2 c1t + c2t +1 + = wt +1 1 + rt + 2
↓
max
U ( c1t , c 2 t +1 , c3 t + 2 ) = u ( c1t ) +
1 1 u ( c 2 t +1 ) + u ( c3 t + 2 ) 2 1+ ρ (1 + ρ )
与P7
↓公式比较
工作期： 0 < θ < 1 ，即相对来说家庭较不愿意改变消 费量在一生各个时期的分配时，个人的消 费和效用水平下降 θ > 1，即相对来说家庭较偏好于波动的消费 流，个人的消费和效用水平提高
退休期： 个人的消费和效用水平得到了提高
二.降低养老金发放水平与个人 福利水平、 福利水平、初始资本
1
c3t + 2
wt +1 1 + rt + 2 θ = [ ] 1 1 2 (1 + ρ ) 1 + rt + 2 θ 1 θ 1 1+ ( ) +[ ] ⋅ 2 1+ ρ (1 + ρ ) 1 + rt + 2
1
1 + rt + 2 θ wt +1 [ ] ⋅ 2 (1 + ρ ) 1 + rt + 2 Tt +1 = wt +1 − c1t − c2t +1 = 1 1 1 + rt + 2 θ 1 θ 1 1+ ( ) +[ ] ⋅ 2 1+ ρ (1 + ρ ) 1 + rt + 2
其中 T = k − kG
*
↓
n − rt + 2 T + wt +1 1 + rt + 2 c1t = 1 1 1 + rt + 2 θ 1 θ 1 1+ ( ) +[ ] ⋅ 2 1+ ρ (1 + ρ ) 1 + rt + 2
n − rt + 2 T + wt +1 1 1 + rt + 2 1 θ c2t +1 = ( ) 1 1 1+ ρ 1 + rt + 2 θ 1 θ 1 1+ ( ) +[ ] ⋅ 2 1+ ρ (1 + ρ ) 1 + rt + 2
Tt + 2 + wt +1 − Tt +1 1 1 + rt + 2 1 θ ( ) c2t +1 = 1 1 1+ ρ 1 + rt + 2 θ 1 θ 1 1+ ( ) +[ ] ⋅ 2 1+ ρ (1 + ρ ) 1 + rt + 2
Tt + 2 + wt +1 − Tt +1 1 1 + rt + 2 1 + rt + 2 θ [ ] = 1 1 2 (1 + ρ ) 1 + rt + 2 θ 1 θ 1 1+ ( ) +[ ] ⋅ 2 1+ ρ (1 + ρ ) 1 + rt + 2
2 2
r * = f ′( k * ) t; kG
*
即经济中出现过度储蓄，居民没有最大化 其两阶段的消费水平和效用水平。
↓ 在完全基金制养老保险的情况下，可能会 存在动态无效率的情况，即个人的消费和 效用水平没有实现最大化，资源的跨时配 置也没有达到最优
如何改善经济的动态效率, 使资源配 置达到帕累托最优呢
↓ 现收现付 ：虽然养老保险的初衷是为了调 节收入分配，以保证个人退休后的最低收 入水平，但其同时可以通过调节个人收入 来影响储蓄，进而影响资本积累。
c1t = K t
c2t +1 = wt +1 − T − Kt − St +1
c3t + 2 = (1 + n)T + (1 + rt + 2 ) St +1
c3t + 2 n − rt + 2 T + wt +1 1 1 + rt + 2 1 + rt + 2 θ [ ] = 1 1 2 (1 + ρ ) 1 + rt + 2 θ 1 θ 1 1+ ( ) +[ ] ⋅ 2 1+ ρ (1 + ρ ) 1 + rt + 2