具有最优工作年限与闲暇的生命周期模型

具有最优工作年限与闲暇的生命周期模型 The Life-Cycle Model with Optimal Years of Working and Leisure first-author 蔡东汉, 男, 教授, 现从事数理经济方面的研究. E-mail: dhcai@whu.edu.cn 蔡东汉, 男, 教授, 现从事数理经济方面的研究. E-mail: dhcai@whu.edu.cn dhcai@whu.edu.cn 武汉大学 数学与统计学院, 湖北 武汉, 430072 武汉大学 数学与统计学院, 湖北 武汉, 430072 College of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, China College of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, China 针对给定生命年限与工资增长率下个人最优工作年限的选择问题, 建立了一个具有消费与闲暇效用的生命周期模型.证明了在给定的条件下个人存在一个最优工作年限使得终身效用最大化, 得出了个人的最优消费与资产路径.证明随着人的寿命延长或工资增长率上升, 个人会延长工作年限.数值仿真结果表明, 个人寿命的延长和工资增长率的上升对最优退休年龄有较大的影响. In this paper, a life-cycle model with utility from both consumption and leisure which is obtained from retirement is setup to inquire how the individuals optimally chose the retiring age facing the given wage growth rate, longevity. It is proved that there exists optimal retiring age under given conditions and the individual's consumption and asset path are given. With individual's longevity prolonging or the rate of wage growth promoting, the individual chooses work longer. By numerical simulation, the optimal retirement is much affected by longevity and wage growth rate. 生命周期模型 最优工作年限 最优消费与资产路径 life-cycle model optimal retirement age optimal consumption and asset path 引言 1 在过去一百年期间最显著的经济变化之一就是退休已经成为个体生命中的重要阶段. Lee [ 1 1 ] 研究了在1850年到1990年期间对每个劳动力市场群体的退休预期时间，这一期间退休后的生存时间增加了6倍多. 在过去的数十年间, 对退休年龄的讨论已经成为经济学家研究的主要问题之一.Chang [ 2 2 ] 认为生存可能性的变化类似于跨时期价格的变化, 预期寿命增加对退休年龄的影响是模棱两可的, 这就意味着决定早退休还是晚退休依赖于老年人的储蓄行为.用关于个人活到75岁主观生存率的数据作为对他们预期寿命的代替, Bloom等人 [ 3 3 ] 的研究表明主观生存率的增加导致了夫妻之间家庭财富的增加, 对工作时间的长度没有影响. Weil [ 4 4 ] 认为退休年龄的增加加重了政府的负担. Hansen和Lonstrup [ 5 5 ] 的研究表明当预期寿命增加时, 享受更多的教育和更早的退休对个体来说是最优的.然而, Echevarria [ 6 6 ] 发现预期寿命的增加可以引起退休年龄和教育年限增加. d’Albis, Sau-Him和Sanchez-Romero [ 7 7 ] 通过对函数进行Voltera微分得出老年人死亡率下降会延迟退休年龄, 但年轻人死亡率的下降可能导致更早的退休, 这是由于人类财富大幅增加了个体的预期寿命.Costa [ 8 8 ] 认为不断增加的终生收入让个体在生命的末期最优地选择更长时间的闲暇. 另一方面, 生命周期模型也引起了经济学家的重视 [ 9 9 ~ 11 11 9~11 ] .如Diamond的两期生命周期模型 [ 12 12 ] ，Blanchard [ 13 13 ] 和Weil [ 14 14 ] 连续生命周期模型，以及Yang等的具有最优教育年限的生命周期模型 [ 15 15 ] ，这些模型分别研究了个人在一生中的消费与储蓄行为及对经济增长的影响. Modigliani和Brumberg [ 16 16 ] 着重研究了退休和终生收入的关系，个人会把其消费均匀地平摊在工作和退休的年份. Ando和Modigliani [ 17 17 ] 总结了Modigliani和Brumberg的关于生命周期模型的假设, 给出了经验分析的验证. Kalemli-Ozcan和Weil [ 18 18 ] 给出一个生命周期模型探讨个人在一生中对工作年限和闲暇进行选择.文中讨论了确定和不确定的死亡日期两种情况.然而, 文中的个人在退休后获得闲暇产生的效用是一个固定的常数, 消费的效用函数是对数效用的形式，且没有给出个人存在最优的退休年龄和最优的消费与资产路径严格的数学证明. 本文中, 个人消费的效用函数假定为常数替代弹性效用函数, 个人从闲暇得到的效用假定为随年龄增加而增大.具有消费与闲暇的个人消费函数是强可加的, 死亡时间是确定的.通过引入工作时间的控制变量, 得到个人面临工作年限与闲暇的最优选择的问题.解优化问题, 文中证明个人存在一个最优工作年限, 并给出个人的最优消费与储蓄路径.探讨了寿命延长和工资增长率上升对个人选择工作年限的影响, 证明随着个人寿命延长或工资增长率上升, 个人会选择工作更长的时间.本文进行的数值仿真表明, 个人寿命的延长和工资增长率的上升对最优退休年龄有较大的影响. 模型的建立 1 文中假定 c ( t ), a ( t ), w ( t )为个人 t 时刻的消费、资产与工资.个人的寿命为 Ω .假定个人出生和死亡时没有资产, 即满足 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459584&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459584&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459584&type=middle 若个人在 t 时刻提供的劳动为 χ ( t ), 0≤ χ ( t )≤1, t ∈[0, Ω ], r 为利率, 则个人的预算约束为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459588&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459588&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459588&type=middle 设个人从消费与闲暇得到效用是强可加的, 为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459591&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459591&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459591&type=middle 其中, v ( t )为个人从闲暇得到的效用, 满足 v (·):[0, Ω ]→[0, ∞), v ′(·)＞0. 于是, 个人的终身效用最大化为在约束条件(1)和(2)下最大化 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459593&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459593&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459593&type=middle 其中, ρ 为时间偏好. 最优工作年限、消费与资产路径的存在性 1 解优化问题的Lagrange函数为 [ 19 19 ] : http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459595&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459595&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459595&type=middle 一阶条件为: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459598&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459598&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459598&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459602&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459602&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459602&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459606&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459606&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459606&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459610&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459610&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459610&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459614&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459614&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459614&type=middle 由(8)式, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459715&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459715&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459715&type=middle .再由(7)式, 得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459620&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459620&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459620&type=middle 即有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459623&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459623&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459623&type=middle 于是, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459627&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459627&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459627&type=middle 其中, 初值 c 0 ( χ ( t ))表示消费函数依赖于控制变量 χ ( t ). 由(1)和(6)式得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459631&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459631&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459631&type=middle 由此, 对给定的 χ ( t ), http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459635&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459635&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459635&type=middle 又由(8)~(10)式, 若 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459639&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459639&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459639&type=middle 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459643&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459643&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459643&type=middle 若 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459647&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459647&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459647&type=middle 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459651&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459651&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459651&type=middle 且 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459655&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459655&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459655&type=middle 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459659&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459659&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459659&type=middle 对给定的 χ ( t ), 记 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459662&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459662&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459662&type=middle 则有以下的引理： 引理1   若 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459718&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459718&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459718&type=middle , 则函数 g ( t )存在惟一的零点 R . 证   由 g (0)=－ w (0)＜0, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459721&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459721&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459721&type=middle , 函数 g ( t )在区间(0, Ω )上至少有一个零点. 令 g 1 ( t )= c 1 ( t ) v ( t ), 则有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459665&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459665&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459665&type=middle http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459667&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459667&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459667&type=middle 在函数 g ( t )的零点 t 1 处有 g 1 ( t 1 )= w ( t 1 ).故 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459670&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459670&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459670&type=middle 这意味着 g ( t )的零点惟一.否则, 设 t 1 , t 2 , t 1 ＜ t 2 为函数 g ( t )的两个邻近的零点, 则存在 ε ＞0使得 t 1 + ε ＜ t 2 － ε 且 g ( t 1 + ε )＞0, g ( t 2 － ε )＜0.由介值定理, 函数在区间( t 1 , t 2 )还存在零点 t 3 .矛盾.引理证毕. 引理2   在引理1的条件下, 控制变量 χ ( t )满足 χ ( t )=1, 0≤ t ＜ R , χ ( t )=0, R ＜ t ≤ Ω . 证   由引理1的证明得, g ( t )＞0, t ∈[0, R )且 g ( t )＜0, t ∈( R , Ω ].故引理成立. 由引理1与引理2, 得到下面的定理： 定理1   在引理1条件下, 个人存在最优工作年限 R . 定理2   个人的最优消费路径为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459674&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459674&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459674&type=middle 定理3   个人的最优资产积累路径为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459677&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459677&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459677&type=middle 寿命与工资增长率对最优工作年限的影响 1 本节假定工资增长率为 γ , 即 w ( t )= w 0 e γ t .记 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459724&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459724&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459724&type=middle 为 c 0 ( R , Ω , r , γ )且令 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459679&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459679&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459679&type=middle 则由 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459681&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459681&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459681&type=middle 及引理1证明中得出的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459727&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459727&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459727&type=middle , 我们得到下面定理： 定理4   方程 Q 1 ( R , Ω )=0确定一个函数 R = R ( Ω )且 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459731&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459731&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459731&type=middle , 这表明个人的最优工作年龄随寿命延长而增加. 令 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459683&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459683&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459683&type=middle 则由 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459686&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459686&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459686&type=middle 及 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459689&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459689&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459689&type=middle 得到下面定理 定理5   当 σ ≥1时, 方程 Q 2 ( R , γ )=0确定一个函数 R = R ( γ )且 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459736&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459736&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459736&type=middle , 这表明个人最优工作年限随工资增长率上升而延长. 证    Q 2 ( R , γ )=0表明 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459690&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459690&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459690&type=middle 由(24)式, 知 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459694&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459694&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459694&type=middle 又由 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459698&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459698&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459698&type=middle 及引理1证明中得出的 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459740&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459740&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459740&type=middle , 定理成立. 数值仿真 1 本节假定闲暇的效用函数为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459744&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459744&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459744&type=middle , 在给定的一组参数下考察个人最大寿命与工资增长率对最优退休年龄的影响.模型中的结构参数取为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459703&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459703&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459703&type=middle 从 图 1 图 1 和 图 2 图 2 中我们看到个人寿命和工资增长率对个人的退休年龄的选择具有较大的影响.在 图 1 图 1 中, 随着个人寿命由80岁增加到100岁, 最优退休年龄由60岁增加到77.4岁; 在 图 2 图 2 中, 随着工资增长率 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459748&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459748&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459748&type=middle 由0.03增加到0.04, 最优退休年龄由60岁增加到68.3岁. 结论 1 本文通过在生命周期模型中引入工作时间的控制变量及常数替代弹性个人消费效用函数与随年龄增加闲暇效用函数, 得到个人终身面临的消费与闲暇最优选择问题.利用解优化问题的一阶条件, 证明个人存在一个最优的工作年限, 并由此给出了个人的最优消费路径和资产积累路径. 文中利用隐函数定理证明, 最优工作年限为人的寿命的单调函数.这一结论表明, 随着人的寿命延长个人选择工作更长的时间; 当工资按指数增长时, 最优工作年限在 σ ＞1时随工资增长率上升会延长. 数值仿真表明, 个人寿命的延长和工资增长率的上升对个人的最优退休年龄的选择有较大的影响.当个人的寿命增加时, 为了确保退休后的消费, 个人需要工作更长的时间; 而当工资增长率上升时, 工作期间可以获得更多的收入, 进而增加整个生命期的消费, 提高了总的效用, 个人也选择更长的工作时间. 参考文献 LEE C. 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Dynamic Optimization: The Calculus of Variations and Optimal Control in Management [M]. 2nd Ed. Amsterdam: Elsevier Science, 1991. 10.14188/j.1671-8836.2017.06.011 O232 whdxxblxb-63-6-533 国家自然科学基金资助项目(71271158) 2017-01-01 2017-12-24 2020-12-09 版权所有© 《武汉大学学报(理学版)》编辑部2017 Copyright © 2017 Journal of Wuhan University(Natural Science Edition). All rights reserved. 2017 武汉大学学报(理学版) 6 63