The Scale Preference of the Residents' Sense of Place and its Factors
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摘要: 以南京农业大学学生为被试，通过半结构化的调查问卷，结合定量和质性分析方法，了解现有行政区划体系下居民对地方依附和地方认同的偏好尺度，分析其人口统计学差异，揭示地方感形成机制与居民地方感偏好尺度的关联.研究结果表明：(1)居民地方依附的尺度偏好呈“倒U型曲线”；地方认同的尺度偏好随尺度递减而下降. (2)年龄、独生子女与否、民族、户籍、专业、入学前的住校时间都会影响居民地方感尺度偏好. (3)地方依附和地方认同作为地方感的从属概念，均包含了情感、认知和行为的过程，都可以采用认同领域的指导性理论加以诠释.Abstract: The students from Nanjing Agricultural University were enrolled as the subjects and data were acquired through a semi-structured questionnaire. With a combination of the qualitative and quantitative methods, data were analyzed and employed to understand the scale preference of the residents' place attachment and place identity under the present administrative division system. Furthermore, the difference of demographic variables in scale distributions of the two dimensions of sense of place (SOP) was examined and the impact of the formation mechanism of SOP on scale preference was discussed. The obtained conclusions are listed as follows. First, the scale distribution of the residents' place attachment is an inverted U-shaped curve in which the medium scales including the city and the county are most attached to while the larger scale of the province and the smaller scales such as the village and the town are much less attached to. The scale distribution of the residents' place identity is different in that the larger the scale, the more it is preferred by the subjects. Second, demographic factors including age, only-childhood, nationality, household registration, university major and length of time living in a dormitory before entering university significantly influence the residents' scale preference of SOP, either on one dimension or both. Third, as the subordinate concepts of SOP, place attachment and place identity both contain the three psychological aspects of feeling, cognition and behavior. In addition, both dimensions can be interpreted with the guiding principles of identity theory, i.e., distinctiveness, continuity, self-efficacy and self-esteem.
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Keywords:
- sense of place /
- scale /
- factors /
- residents
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随着公司股份制改革, 为了保证公司现金流的充足和运营安全, 公司在发行股票筹集资金时需考虑回馈作为投资者的股东, 即分红.分红问题中, 讨论得较多的有经典风险模型和风险对偶模型[1-6], 其中风险对偶模型可描述为[1]:
U(t)=u−ct+S(t)(t∈N+,u∈N), (1) 其中, U(0)=u表示公司的初始资金, 正整数c表示在单位时间(t-1, t]内的支出, S(t)表示直到t时刻的总收益.
风险对偶模型中研究得较多的分红策略有:Barrier策略[1]和Threshold策略[2], 例如:运用Laplace变换方法讨论了复合Poisson对偶模型的最优分红Barrier的确定方法[1]; 利用2个Integro积分方程和Laplace变换给出了最优分红Threshold的计算方法[2].显然, 在最优分红问题中, 既提高股东收益又降低公司风险的方案应当同时考虑分红与再注资[7-13], 例如:在离散经典风险模型中证明了最优值函数是一个Hamilton-Jacobi-Bellman(HJB)方程的唯一解, 指出再注资后最优分红策略是Barrier策略[7]; 在对偶模型中证明了带注资的最优分红策略为Barrier策略[8]; 讨论了带比例和固定交易费的再注资的最优分红问题, 并通过数值实例说明分红边界随交易费比例的增加而上升[9]; 在复合二项风险模型中证明了最优值函数是一个HJB方程的唯一解, 并验证了最优控制策略是双Barrier策略[10].为了更贴合实际, 考虑分红贴现利率的变化具有随机性[14-15], 例如:在随机利率下讨论了离散风险模型中具有延迟索赔的最优分红问题, 得到了最优策略的一个高效算法[14].
本文在红利有界的条件下, 研究复合二项对偶模型中带比例交易费再注资且分红贴现利率随机变化的最优分红问题; 运用压缩映射不动点原理证明了该最优分红问题的最优值函数是一个离散HJB方程的唯一解, 得到了最优分红策略和最优值函数的计算方法; 为了能在实际运用中计算最优红利值, 根据分红策略的一些性质得到了该最优值函数的可无限逼近的上界和下界; 最后给出数值实例来验证本文所给的最优分红策略的有效性.
1. 预备知识
文中用到的相关记号、符号如下:
(ⅰ) N={0, 1, 2, …}; N+={1, 2, …}; Nk={0, 1, 2, …, k} (k∈N).
(ⅱ) c, C, m, x∈N+; i, j=1, 2, …, m.
(ⅲ) Ft是σ代数, 包含了t时刻及之前的所有信息.
(ⅳ) a1 ∨ a2=max{a1, a2}, a1 ∧ a2=min{a1, a2}.
假设任意单位时间(t-1, t] (t∈N+)内至多有一次收入, 在t的前一瞬时结算.用εt=1表示有一次收入，收入量为Xt∈N+; εt=0表示无收入.序列{Xt}和{εt}分别为独立同分布的随机变量, 其中{εt}具有概率Pr(εt=1)=p (0 < p < 1), Pr(εt=0)=q=1-p, {Xt}具有概率函数f(x)=Pr(Xt=x)和分布函数F(x)=x∑l=1f(l), 且与{εt}相互独立.则直到时刻t的总收益S(t)是复合二项序列[16]:
S(t)={0(t=0),X1ε1+X2ε2+⋯+Xtεt(t≥1). 再假设(t-1, t]时间段的利率{Rt, t∈N+}是有限状态空间{r1, r2, …, rm}的关于Ft可测的齐次Markov链, 一步转移概率阵为P=(pij)i, j=1m, 其中pij=Pr(Rt+1=rj|Rt=ri).令vi (0 < vi < 1)为第i时间段内的贴现因子, v=max{v1, v2, …, vm}, 则vi=1/(1+ri).
在模型(1)中引入分红策略.假设在0时刻不考虑分红, 分别用dt和zt表示t(t∈N)时刻的分红和再注资, 记d0=0.在任何时刻t的分红策略满足以下4个条件称为可行的策略:(1)dt取整数且有上界C; (2)dt和zt关于Ft可测; (3)盈余u不大于c时不分红, 并由股东注入相应的资金c-u(存在交易费), 使盈余能够快速恢复到c; (4)盈余u不小于c时不注资, 分红由超出c的部分承担.
由于模型(1)具有Markov性质, 所以只需讨论依赖于各时刻盈余的可行策略, 这类策略是关于盈余的函数, 组成的集合用Λ表示.为方便讨论, 用φi(u)表示盈余为u、利率状态为ri时的分红量, 则任意时刻t (t∈N+)的利率Rt=rj时的分红量为φj(U(t-1)-c+Xtεt).故初始利率为R0=ri的累积分红折现均值函数(以下简称值函数)[7]为
Vi(u;φi)=E[∞∑k=1dk(k∏t=111+Rt)−∞∑k=0zk(k∏t=111+Rt)|R0=ri], (2) 记0∏t=111+Rt=1.
优化目标是找到最优值函数
V∗i(u;φi)=supφi∈ΛVi(u;φi) 和对应的最优分红策略, 记为φi*(u)=argmax Vi(u, φi).为方便讨论, Vi(u; φi)记为Vi(u).
2. 最优分红策略和最优值函数
对任意的φi(u)∈Λ, 根据全期望公式[17], 带比例交易费再注资的值函数Vi(u)满足:
Vi(u)={(1+β)(u−c)+Vi(c)(0≤u<c),qm∑j=1pijvj[Vj(u−c−φj(u−c))+φj(u−c)]+pm∑j=1pijvj∞∑x=1[Vj(u−c+x−φj(u−c+x))φj(u−c+x)]f(x)(u≥c), (3) 其中β(β≥0)为交易费的比例(常量).
定理1 对任意的φi(u)∈Λ, 最优值函数Vi*(u)满足如下HJB方程:
V∗i(u)={(1+β)(u−c)+V∗i(c)(0⩽u<c),qm∑j=1pijvj[maxa∈N(C∧(u−2c))∨0(Vj(u−c−a)+a)]+pm∑j=1pijvj∞∑x=1[maxa∈N(C∧(u−2c+x))∨0(Vj(u−c+x−a)+a)f(x)](u⩾c). (4) 证明 对式(3)取最优值得:
V∗i(u)={supφi(u)∈Λ{(1+β)(u−c)+Vi(c)}(0≤u<c),supφi(u)∈Λ{qm∑j=1pijvj[Vj(u−c−φj(u−c))+φj(u−c)]+pm∑j=1pijvj∞∑x=1[Vj(u−c+x−φj(u−c+x))+φj(u−c+x)]f(x)}(u≥c). (5) 显然式(5)与式(4)等价.证毕.
定义1[16] 记H表示所有m维有界实序列组成的集合.对H中任意两点X=(Xi(u))和Y=(Yi(u))(u∈N, i=1, 2, …, m), 称
d\left( {\mathit{\boldsymbol{X}},\mathit{\boldsymbol{Y}}} \right) = \left\| {\mathit{\boldsymbol{X}} - \mathit{\boldsymbol{Y}}} \right\| = \mathop {\sup }\limits_{u,i} \left| {{X_i}\left( u \right) - {Y_i}\left( u \right)} \right| 为X、Y的距离.显然H=(H, d)是完备度量空间, 且任意的V =(Vi(u))∈H.
对任意的V =(Vi(u))∈H (i={1, 2, …, m}), 定义
\begin{array}{*{20}{l}} {b_{{V_i}}}\left( u \right) = \left\{ \begin{array}{*{20}{l}} 0\;\;\;\;\left( {0 \leqslant u \leqslant c} \right), \hfill \\ \arg \mathop {\max }\limits_{d \in {\mathbb{N}_{C \wedge \left( {u - c} \right)}}} \left\{ {{V_i}\left( {u - d} \right) + d} \right\}\;\;\;\;\;\left( {u > c} \right). \hfill \\ \end{array} \right. \end{array} 又因式(4)中当0≤u < c时, Vi*(u)的解的情况仅依赖于Vi*(c)的解, 下面讨论u≥c时Vi*(u)的解的情况.
记T =(T1, T2, …, Tm), 其中Ti (i=1, 2, …, m)为H上的m个算子, 满足:
\begin{array}{l} {T_i}\mathit{\boldsymbol{V}} = q\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\left[ {{V_j}\left( {u - c - {b_{{V_j}}}\left( {u - c} \right)} \right) + {b_{{V_j}}}\left( {u - c} \right)} \right] + \\ \;\;\;\;\;\;\;\;p\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\sum\limits_{x = 1}^\infty {\left[ {{V_j}\left( {u - c + x - {b_{{V_j}}}\left( {u - c + x} \right)} \right) + } \right.} \\ \;\;\;\;\;\;\;\;\left. {{b_{{V_j}}}\left( {u - c + x} \right)} \right]f\left( x \right)\;\;\;\;\left( {u \ge c} \right), \end{array} (6) 或
\mathit{\boldsymbol{TV}} = \left( {{T_1}\mathit{\boldsymbol{V}},{T_2}\mathit{\boldsymbol{V}}, \cdots ,{T_m}\mathit{\boldsymbol{V}}} \right), (7) 则式(4)中u≥c的部分等价于
{T_i}\mathit{\boldsymbol{V}} = {V_i}, (8) 或
\mathit{\boldsymbol{TV}} = \mathit{\boldsymbol{V}}. (9) 定理2 方程(9)有且仅有唯一解.
证明 假设对任意Xi(u), Yi(u)∈H, i={1, 2, …, m}, 及给定的u≥c, Xi(u-bXi(u))+bXi(u)≥Yi(u-bYi(u))+bYi(u), 因为Yi(u-bYi(u))+bYi(u)≥Yi(u-bXi(u))+bXi(u), 所以
\begin{array}{*{20}{l}} \mathop {\sup }\limits_{u \geqslant c} \left| {{X_i}\left( {u - {b_{{X_i}}}\left( u \right)} \right) + {b_{{X_i}}}\left( u \right) - \left[ {{Y_i}\left( {u - {b_{{Y_i}}}\left( u \right)} \right) + {b_{{Y_i}}}\left( u \right)} \right]} \right| \leqslant \hfill \\ \;\;\;\mathop {\sup }\limits_{u \geqslant c} \left| {{X_i}\left( {u - {b_{{X_i}}}\left( u \right)} \right) + {b_{{X_i}}}\left( u \right) - \left[ {{Y_i}\left( {u - {b_{{X_i}}}\left( u \right)} \right) + {b_{{X_i}}}\left( u \right)} \right]} \right| \leqslant \hfill \\ \;\;\;\mathop {\sup }\limits_{u \geqslant c} \left| {{X_i}\left( u \right) - {Y_i}\left( u \right)} \right| \leqslant \left\| {\mathit{\boldsymbol{X}} - \mathit{\boldsymbol{Y}}} \right\|, \hfill \\ \end{array} 则
\begin{gathered} \mathop {\sup }\limits_{u \geqslant c} \left| {{T_i}\mathit{\boldsymbol{X}} - {T_i}\mathit{\boldsymbol{Y}}} \right| \leqslant q\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\left\| {\mathit{\boldsymbol{X}} - \mathit{\boldsymbol{Y}}} \right\| + \hfill \\ \;\;\;\;p\sum\limits_{i = 1}^m {{p_{ij}}} {v_j}\sum\limits_{x = 1}^\infty {\left\| {\mathit{\boldsymbol{X}} - \mathit{\boldsymbol{Y}}} \right\|} f(x) \leqslant v\left\| {\mathit{\boldsymbol{X}} - \mathit{\boldsymbol{Y}}} \right\|. \hfill \\ \end{gathered} 故对任意的u≥c, 有
d\left( {\mathit{\boldsymbol{TX}},\mathit{\boldsymbol{TY}}} \right) \leqslant vd\left( {\mathit{\boldsymbol{X}},\mathit{\boldsymbol{Y}}} \right). 在0 < v < 1的假设下, 算子T是压缩映射, 因此, 当u≥c时, 方程(9)有且仅有唯一解.证毕.
结论1 对任意u∈ \mathbb{N} , i={1, 2, …, m}, 方程(4)有且仅有唯一解, 且该唯一解为最优值函数Vi*(u).
定理3 对任意的φi(u)∈Λ, u∈ \mathbb{N} , i={1, 2, …, m}, Vi(u)取最优值时当且仅当
\begin{array}{*{20}{l}} \varphi _i^*\left( u \right) = \left\{ {\begin{array}{*{20}{l}} 0\ \ \ \ \ {(0 \leqslant u \leqslant c),} \\ {\arg \mathop {\max }\limits_{_{d \in {\mathbb{N}_{C \wedge (u - c)}}}} \left( {{V_i}(u - d) + d} \right)}&{(u > c).} \end{array}} \right. \end{array} (10) 证明 (1)必要性.由模型的假设知, 当0≤u≤c时, φi(u)=0.对任意的i={1, 2, …, m}, 若Vi(u)是最优的值函数, 则Vi(u)满足式(3)、(4).比较式(3)与式(4), 当u>c时, 有
{V_i}\left( {u - {\varphi _i}\left( u \right)} \right) + {\varphi _i}\left( u \right) = \mathop {\max }\limits_{_{d \in {\mathbb{N}_{C \wedge (u - c)}}}} \left( {{V_i}(u - d) + d} \right), 因此式(10)成立.
(2) 充分性.因方程(4)的解存在且唯一, 故充分性成立.证毕.
3. 最优红利值的算法
采用Bellman递归算法, 计算最优值函数列(Vi*(u))和最优分红策略φi*(u) (u∈ \mathbb{N} )的值[16].即任意给定一个初始函数列(Vi(0)(u)), 根据下式计算(Vi(s)(u)) (s=1, 2, …):
\begin{array}{*{20}{l}} V_i^{(s)}(u) = \hfill \\ \left\{ \begin{array}{*{20}{l}} \left( {1 + \beta } \right)\left( {u - c} \right) + V_i^{\left( {s - 1} \right)}(c)\;\;\;\;\left( {0 \leqslant u < c} \right), \hfill \\ q\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\left[ {V_j^{(s - 1)}\left( {u - c - {b_{V_j^{\left( {s - 1} \right)}}}\left( {u - c} \right)} \right) + {b_{V_j^{\left( {s - 1} \right)}}}\left( {u - c} \right)} \right] + \hfill \\ p\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\sum\limits_{x = 1}^\infty {\left[ {V_j^{(s - 1)}\left( {u - c + x - {b_{V_j^{(s - 1)}}}(u - c + x)} \right) + } \right.} \hfill \\ \left. {{b_{V_j^{(s - 1)}}}(u - c + x)} \right]f(x)\;\;\;\;(u \geqslant c). \hfill \\ \end{array} \right. \hfill \\ \end{array} (11) 由不动点原理及式(8)、(11)得
V_i^ * \left( u \right) = \mathop {\lim }\limits_{n \to \infty } V_i^{(n)}(u) = \mathop {\lim }\limits_{n \to \infty } T_i^nV_i^{(0)}(u), n充分大时可以用Vi(n)(u)近似Vi*(u), 它们之间的误差估计为
d\left( {V_i^{(n)},V_i^*} \right) \leqslant \frac{{{v^n}}}{{1 - v}}. (12) 当时间区间(t-1, t]内可能的收入不是有界的随机变量时, 递归式(11)中的函数序列存在无穷项之和, 不便于数值计算.在C < ∞的情形下, 对任意的n1∈ \mathbb{N} +, 考虑如下2个方程:
\begin{array}{*{20}{l}} V_i^{*\left( 1 \right)}\left( u \right) = \hfill \\ \;\;\;\left\{ \begin{array}{*{20}{l}} (1 + \beta )(u - c) + V_i^{*(1)}(c)\quad (0 \leqslant u < c), \hfill \\ q\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\left[ {V_j^{*(1)}\left( {u - c - {b_{V_j^*}}(u - c)} \right) + {b_{V_j^*(1)}}(u - c)} \right] + \hfill \\ \;\;\;p\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\sum\limits_{x = 1}^{{n_1}} {\left[ {V_j^{*(1)}\left( {u - c + x - {b_{V_j^*(1)}}(u - c + x)} \right) + } \right.} \hfill \\ \left. {{b_{V_j^*(1)}}(u - c + x)} \right]f(x)\;\;\;\;\;(u \geqslant c); \hfill \\ \end{array} \right. \hfill \\ \end{array} (13) \begin{array}{*{20}{l}} V_i^{ * \left( 2 \right)}\left( u \right) = \hfill \\ \;\;\;\left\{ \begin{array}{*{20}{l}} (1 + \beta )(u - c) + V_i^{*(2)}(c)\;\;\;\;(0 \leqslant u < c), \hfill \\ q\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\left[ {V_j^{*(2)}\left( {u - c - {b_{V_j^{*(2)}}}(u - c)} \right) + {b_{V_j^{*(2)}}}(u - c)} \right] + \hfill \\ \;\;\;\;p\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\sum\limits_{x = 1}^{{n_1}} {\left[ {V_j^{*(2)}\left( {u - c + x - {b_{V_j^*(2)}}(u - c + x)} \right) + } \right.} \hfill \\ \left. {\;\;\;\;\;{b_{{V_j}*(2)}}(u - c + x)} \right]f(x) + \frac{{pvC}}{{1 - v}}\bar F\left( {{n_1}} \right)\quad (u \geqslant c). \hfill \\ \end{array} \right. \hfill \\ \end{array} (14) 定理4 对任意的u ∈ \mathbb{N} , i={1, 2, …, m}, 有
V_i^{*(1)}(u) \leqslant V_i^*(u) \leqslant V_i^{*(2)}(u). (15) 证明 (1)先证Vi*(1)(u)≤Vi*(u) (u≥c).
对任意的u≥c和给定的i, 记Wi(1)(u)为方程(6)等号的右边, 则Vi*(1)(u)≤Wi(1)(u)≤TiWi(1)(u).记Wi(2)(u)=TiWi(1)(u), 则Wi(1)(u)≤Wi(2)(u)≤TiWi(2)(u).依次类推, 可得一个递增的函数序列{Wi(n2)(u), n2=1, 2, …}满足Wi(n2+1)=TiWi(n2), 有
\mathop {\lim }\limits_{{n_2} \to \infty } W_i^{\left( {{n_2}} \right)}(u) = V_i^*(u). 因此,
V_i^{*(1)}(u) \leqslant V_i^*(u)\quad (u \geqslant c) (2) 同理可证Vi*(u)≤Vi(2)(u) (u≥c)[18].
(3) 由证明过程(1)、(2), 当u≥c时, 有Vi*(1)(u)≤Vi*(u)≤Vi*(2)(u); 由式(4)、(13)、(14), 当0≤u < c时, 有Vi*(1)(u)≤Vi*(u)≤Vi*(2)(u).证毕.
推论1 对任意的0 < v < 1, n1, u∈ \mathbb{N} , i={1, 2, …, m}, 有
d\left( {V_i^{*(1)},V_i^{*(2)}} \right) \leqslant \frac{{pvC\bar F\left( {{n_1}} \right)}}{{(1 - v)\left( {1 - qv - pvF\left( {{n_1}} \right)} \right)}}. (16) 证明 由式(13)、(14), 当0≤u < c时, 有
d\left( {V_i^{*(1)}(u),V_i^{*(2)}(u)} \right) \leqslant d\left( {V_i^{*(1)}(c),V_i^{*(2)}(c)} \right). 当u≥c时, 由定理2的证明过程，有
\begin{gathered} d\left( {V_i^{*(1)},V_i^{*(2)}} \right) \leqslant q\sum\limits_{i = 1}^m {{p_{ij}}} {v_j}d\left( {V_i^{*(1)},V_i^{*(2)}} \right) + \hfill \\ \;\;\;\;\;p\sum\limits_{i = 1}^m {{p_{ij}}} {v_j}\sum\limits_{x = 1}^{{n_1}} d \left( {V_i^{*(1)},V_i^{*(2)}} \right)f(x) + \frac{{pvC}}{{1 - v}}\bar F\left( {{n_1}} \right) \leqslant \hfill \\ \;\;\;\;\;\left( {qv + pvF\left( {{n_1}} \right)} \right)d\left( {V_i^{*(1)},V_i^{*(2)}} \right) + \frac{{pvC}}{{1 - v}}\bar F\left( {{n_1}} \right), \hfill \\ \end{gathered} 则式(16)成立.证毕.
当n1充分大时, Vi*(1)(u)和Vi*(2)(u)能无限逼近最优值函数Vi*(u).定义算子\widetilde{\boldsymbol{T}}=\left(\tilde{T}_{1}, \tilde{T}_{2}, \cdots\right., \left.\tilde{T}_{m}\right), 其中\widetilde{\boldsymbol{T}} V_{i}^{*(1)}(u)等于式(13)等号的右边.显然, \tilde{\boldsymbol{T}}也是H上的压缩映射.运用式(13)或式(14)变换迭代公式(11)后，可进行数值实例的计算.
综上可得最优红利值的算法:
第1步, 精度控制.给定一个精度要求, 根据式(12)得到迭代步数n及式(16)中n1的值.
第2步, 迭代计算.任意给定一个初始函数列(Vi(0)(u)) (0≤u≤n1(n+1)), 由式(13)或式(14)变换迭代公式(11)，然后计算(Vi(s)(u)) (0≤u≤n1(n-s+1)), 其中s={1, 2, …, n}, 最终得到φi*(u)和Vi*(u)的近似值.
4. 数值模拟
本节列举了一个几何分布的实例, 利用第3节提出的最优红利值的算法来计算函数φi*(u)和Vi*(u), 其中转移概率矩阵采用与文献[14]、[19]类似的方法构造.为进一步对比分析, 计算不注资时的最优分红策略和最优值函数, 满足条件:(1)分红有上界, 且任何时刻的分红都不超过该时刻的盈余; (2)当盈余为负值时公司破产.不注资时的最优值函数Wi*(u)为[16]:
\begin{array}{*{20}{l}} W_i^*(u) = q\sum\limits_{j = 1}^m {{p_{ij}}} {v_j}\left( {W_j^*\left( {u - c - {b_{W_j^*}}(u - c)} \right) + {b_{W_j^*}}(u - c)} \right) + \hfill \\ \;\;\;\;\;\;\;\;p\sum\limits_{j = 1}^\infty {{p_{ij}}} {v_j}\sum\limits_{x = 1}^\infty {\left( {W_j^*\left( {u - c + x - {b_{W_j^*}}(u - c + x)} \right) + } \right.} \hfill \\ \;\;\;\;\;\;\;\;\left. {{b_{W_j^*}}(u - c + x)} \right)f(x)\;\;\;\;\;(u \geqslant 0). \hfill \\ \end{array} (17) 例1 假设p=0.7, c=C=5, 收入量服从均值为μ=10的几何分布, 概率函数为:
f(x) = \frac{1}{{10}}{\left( {\frac{9}{{10}}} \right)^{x - 1}}\;\;\;\;\;(x = 1,2, \cdots ). 再假设利率为:r1=0.05, r2=0.04, r3=0.03, 转移概率矩阵为:
\mathit{\boldsymbol{P}} = \left( {\begin{array}{*{20}{c}} {0.5}&{0.2}&{0.3} \\ {0.25}&{0.6}&{0.15} \\ {0.1}&{0.2}&{0.7} \end{array}} \right). 表 1是当交易费比例为β=0, 0.2, 0.4, 0.6, 1.2及不注资时, 最优分红策略φi*(u)对应的值.表 2是当交易费比例为β=0, 0.2, 1.2以及不注资时, 最优值函数Vi*(u)的部分近似值(精度为10-4).
表 1 最优分红策略φi*(u) (i=1, 2, 3, C=5)Table 1. The optimal dividend strategy φi*(u) (i=1, 2, 3, C=5)u β=0 β=0.2 β=0.4 β=0.6 β=1.2 不注资 i=1 i=2 i=3 i=1 i=2 i=3 i=1 i=2 i=3 i=1 i=2 i=3 i=1 i=2 i=3 i=1 i=2 i=3 0, 1, …, 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 12 2 2 2 2 2 2 1 0 0 0 0 0 0 0 0 0 0 0 13 3 3 3 3 3 3 2 1 1 0 0 0 0 0 0 0 0 0 14 4 4 4 4 4 4 3 2 2 0 0 0 0 0 0 0 0 0 15 5 5 5 5 5 5 4 3 3 0 0 0 0 0 0 0 0 0 16 5 5 5 5 5 5 5 4 4 1 1 1 0 0 0 0 0 0 17 5 5 5 5 5 5 5 5 5 2 2 2 0 0 0 0 0 0 18 5 5 5 5 5 5 5 5 5 3 3 3 0 0 0 0 0 0 19 5 5 5 5 5 5 5 5 5 4 4 4 0 0 0 0 0 0 20 5 5 5 5 5 5 5 5 5 5 5 5 0 0 0 0 0 0 21 5 5 5 5 5 5 5 5 5 5 5 5 1 1 1 1 1 1 22 5 5 5 5 5 5 5 5 5 5 5 5 2 2 2 2 2 2 23 5 5 5 5 5 5 5 5 5 5 5 5 3 3 3 3 3 3 24 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 4 4 4 25, 26, … 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 表 2 最优值函数Table 2. The optimal value functionu β=0 β=0.2 β=1.2 不注资 V1*(u) V2*(u) V3*(u) V1*(u) V2*(u) V3*(u) V1*(u) V2*(u) V3*(u) W1*(u) W2*(u) W3*(u) 0 34.650 1 35.061 4 35.033 0 29.566 9 29.938 8 29.912 0 8.898 5 9.126 3 9.102 4 10.566 7 10.673 5 10.667 2 1 35.650 1 36.061 4 36.033 0 30.766 9 31.138 8 31.112 0 11.098 5 11.326 3 11.302 4 11.740 8 11.859 4 11.852 5 2 36.650 1 37.061 4 37.033 0 31.966 9 32.338 8 32.312 0 13.298 5 13.526 3 13.502 4 13.045 4 13.177 2 13.169 4 3 37.650 1 38.061 4 38.033 0 33.166 9 33.538 8 33.512 0 15.498 5 15.726 3 15.702 4 14.494 9 14.641 3 14.632 7 4 38.650 1 39.061 4 39.033 0 34.366 9 34.738 8 34.712 0 17.698 5 17.926 3 17.902 4 16.105 4 16.268 1 16.258 5 5 39.650 1 40.061 4 40.033 0 35.566 9 35.938 8 35.912 0 19.898 5 20.126 3 20.102 4 20.153 1 20.357 0 20.344 9 6 40.563 5 40.981 1 40.953 3 36.611 9 36.990 9 36.964 8 21.621 3 21.861 2 21.838 3 21.514 1 21.731 7 21.718 8 7 41.471 9 41.896 0 41.868 7 37.645 3 38.031 5 38.006 1 23.301 5 23.553 2 23.531 4 22.928 8 23.160 6 23.146 9 8 42.374 9 42.805 5 42.778 7 38.665 7 39.059 2 39.034 4 24.934 2 25.197 8 25.176 8 24.392 3 24.638 7 24.624 2 9 43.271 8 43.708 9 43.682 6 39.671 9 40.072 7 40.048 5 26.514 1 26.789 7 26.769 4 25.897 9 26.159 4 26.144 0 10 44.161 9 44.605 7 44.579 8 40.662 1 41.070 4 41.046 6 28.035 4 28.322 8 28.303 1 27.919 5 28.201 3 28.184 9 11 45.026 6 45.477 8 45.452 1 41.602 2 42.018 6 41.995 0 29.391 2 29.691 0 29.671 3 29.159 1 29.453 2 29.436 1 12 45.888 8 46.347 3 46.321 8 42.532 9 42.957 1 42.933 7 30.704 5 31.016 4 30.996 6 30.402 4 30.708 8 30.691 0 13 46.748 7 47.214 4 47.189 1 43.453 8 43.885 9 43.862 6 31.974 3 32.298 0 32.278 1 31.643 1 31.961 7 31.943 4 14 47.606 4 48.079 3 48.054 2 44.365 1 44.804 8 44.781 8 33.199 6 33.534 7 33.514 7 32.874 3 33.204 9 33.186 1 15 48.462 3 48.942 3 48.917 3 45.266 9 45.714 1 45.691 2 34.380 0 34.726 1 34.705 8 34.191 7 34.535 2 34.515 8 16 49.341 0 49.827 5 49.802 9 46.164 5 46.618 4 46.596 0 35.493 8 35.850 4 35.829 9 35.272 3 35.626 3 35.606 4 17 50.210 9 50.704 0 50.679 8 47.055 5 47.516 2 47.494 1 36.577 4 36.944 3 36.923 6 36.341 4 36.705 5 36.685 3 18 51.071 1 51.570 8 51.546 9 47.939 0 48.406 6 48.384 8 37.631 9 38.008 7 37.987 7 37.395 8 37.770 0 37.749 4 19 51.920 3 52.426 9 52.403 2 48.814 2 49.289 0 49.267 4 38.658 4 39.044 8 39.023 6 38.432 8 38.816 7 38.795 8 20 52.757 5 53.271 1 53.247 6 49.680 3 50.162 3 50.140 8 39.658 4 40.054 0 40.032 6 39.471 9 39.865 5 39.844 3 21 53.553 2 54.074 4 54.050 7 50.524 1 51.014 3 50.992 6 40.629 3 41.033 7 41.012 2 40.435 0 40.837 4 40.816 1 22 54.344 3 54.873 2 54.849 3 51.359 2 51.857 5 51.835 6 41.582 6 41.995 6 41.974 0 41.387 3 41.798 3 41.776 8 23 55.130 8 55.667 3 55.643 1 52.185 7 52.692 0 52.669 8 42.519 7 42.941 1 42.919 4 42.328 0 42.747 4 42.725 8 24 55.912 3 56.456 4 56.432 0 53.003 5 53.517 7 53.495 4 43.442 4 43.871 9 43.850 2 43.256 8 43.684 4 43.662 8 25 56.688 7 57.240 3 57.215 7 53.812 9 54.335 0 54.312 3 44.352 3 44.789 6 44.767 9 44.178 5 44.614 0 44.592 5 26 57.464 8 58.023 8 57.999 1 54.614 9 55.144 7 55.121 9 45.255 6 45.699 7 45.678 4 45.082 5 45.524 8 45.503 6 27 58.231 4 58.797 9 58.772 9 55.408 8 55.946 3 55.923 3 46.152 9 46.603 9 46.582 9 45.980 5 46.429 6 46.408 8 28 58.988 2 59.562 1 59.536 9 56.194 2 56.739 4 56.716 1 47.043 5 47.501 4 47.480 7 46.871 8 47.328 0 47.307 4 29 59.735 0 60.316 3 60.290 7 56.970 8 57.523 7 57.500 0 47.926 5 48.391 7 48.371 1 47.755 7 48.219 2 48.198 7 30 60.471 5 61.060 1 61.034 2 57.738 3 58.298 8 58.274 8 48.801 3 49.273 9 49.253 4 48.631 4 49.102 3 49.081 9 由表 1可知:β=0, 0.1, 0.4, 0.6, 1.2时, φi*(u)的门槛值分别为10、10、12、16、20, 表明本例给出的最优分红策略φi*(u)是Threshold策略[2].由表 2知:在资金u相同的条件下, 该最优分红策略φi*(u)所对应的最优值函数Vi*(u)与交易费比例β有关, Vi*(u)的值随着β的增大而减少; 再对比表 2中未注资的最优值函数Wi*(u), 发现交易费比例β控制在一定范围内时Vi*(u)明显大于Wi*(u).这说明了本文的最优红利值算法的可行性, 也验证了本文的最优分红策略的有效性.
5. 小结
本文在复合二项对偶模型中讨论了带比例交易费再注资且分红贴现利率随机变化的最优分红问题, 运用压缩映射不动点原理证明了该最优分红问题的最优值函数是一个离散的HJB方程的唯一解, 得到了最优分红策略和最优值函数的优化算法.数值实例结果表明:当交易费比例控制在一定范围内时，相应的最优红利值大于未注资的最优红利值, 说明了文中最优分红策略的有效性.在今后的工作中, 可进一步研究最优分红策略的Threshold性质, 也可尝试在红利无限制的条件下讨论最优分红问题.
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表 1 被试地方感偏好尺度的频数分布
Table 1 The frequency distribution of scale preference of the subjects' sense of place
次 维度/分组 省/直辖市/自治区 地级市/省辖市/自治州 县/自治县/区 乡/民族乡/镇/街道 村/社区 地方依附 54(20.7%) 89(34.1%) 81(31.0%) 17(6.5%) 20(7.7%) 男生 18(24.0%) 26(34.7%) 18(24.0%) 5(6.7%) 8(10.7%) 女生 36(19.4%) 63(33.9%) 63(33.9%) 12(6.5%) 12(6.5%) 19~20岁 7(14.3%) 15(30.6%) 19(38.8%) 4(8.2%) 4(8.2%) 21岁 18(17.1%) 44(41.9%) 37(35.2%) 4(3.8%) 2(1.9%) 22岁及以上 29(27.1%) 30(28.0%) 25(23.4%) 9(8.4%) 14(13.1%) 独生子女 31(18.7%) 70(42.2%) 48(28.9%) 9(5.4%) 8(4.8%) 非独子女 23(24.2%) 19(20.0%) 33(34.7%) 8(8.4%) 12(12.6%) 汉族 44(18.9%) 87(37.3%) 74(31.8%) 15(6.4%) 13(5.6%) 少数民族 10(35.7%) 2(7.1%) 7(25.0%) 2(7.1%) 7(25.0%) 城规专业 21(21.4%) 23(23.5%) 39(39.8%) 7(7.1%) 8(8.2%) 人力专业 8(14.3%) 28(50.0%) 14(25.0%) 3(5.4%) 3(5.4%) 土管专业 16(21.1%) 28(36.8%) 21(27.6%) 4(5.3%) 7(9.2%) 其他专业 9(29.0%) 10(32.3%) 7(22.6%) 3(9.7%) 2(6.5%) 一年级 1(3.6%) 6(21.4%) 16(57.1%) 1(3.6%) 4(14.3%) 二年级 9(29.0%) 9(29.0%) 8(25.8%) 4(12.9%) 1(3.2%) 三年级 42(21.4%) 73(37.2%) 55(28.1%) 11(5.6%) 15(7.7%) 四年级 2(33.3%) 1(16.7%) 2(33.3%) 1(16.7%) 0(0.0%) 非农户籍 35(21.6%) 69(42.6%) 46(28.4%) 10(6.2%) 2(1.2%) 农业户籍 19(19.2%) 20(20.2%) 35(35.4%) 7(7.1%) 18(18.2%) 未住校 16(17.8%) 32(35.6%) 36(40.0%) 3(3.3%) 3(3.3%) 住校1~3年 22(23.2%) 38(40.0%) 20(21.1%) 9(9.5%) 6(6.3%) 住校4年及以上 16(21.1%) 19(25.0%) 25(32.9%) 5(6.6%) 11(14.5%) 地方认同 146(55.9%) 92(35.2%) 23(8.8%) 0(0.0%) 0(0.0%) 男生 43(57.3%) 28(37.3%) 4(5.3%) 0(0.0%) 0(0.0%) 女生 103(55.4%) 64(34.4%) 19(10.2%) 0(0.0%) 0(0.0%) 19~20岁 26(53.1%) 21(42.9%) 2(4.1%) 0(0.0%) 0(0.0%) 21岁 55(52.4%) 39(37.1%) 11(10.5%) 0(0.0%) 0(0.0%) 22岁及以上 65(60.7%) 32(29.9%) 10(9.3%) 0(0.0%) 0(0.0%) 独生子女 80(48.2%) 73(44.0%) 13(7.8%) 0(0.0%) 0(0.0%) 非独子女 66(69.5%) 19(20.0%) 10(10.5%) 0(0.0%) 0(0.0%) 汉族 124(53.2%) 88(37.8%) 21(9.0%) 0(0.0%) 0(0.0%) 少数民族 22(78.6%) 4(14.3%) 2(7.1%) 0(0.0%) 0(0.0%) 城规专业 53(54.1%) 34(34.7%) 11(11.2%) 0(0.0%) 0(0.0%) 人力专业 33(58.9%) 19(33.9%) 4(7.1%) 0(0.0%) 0(0.0%) 土管专业 40(52.6%) 29(38.2%) 7(9.2%) 0(0.0%) 0(0.0%) 其他专业 20(64.5%) 10(32.3%) 1(3.2%) 0(0.0%) 0(0.0%) 一年级 13(46.4%) 14(50.0%) 1(3.6%) 0(0.0%) 0(0.0%) 二年级 20(64.5%) 8(25.8%) 3(9.7%) 0(0.0%) 0(0.0%) 三年级 109(55.6%) 68(34.7%) 19(9.7%) 0(0.0%) 0(0.0%) 四年级 4(66.7%) 2(33.3%) 0(0.0%) 0(0.0%) 0(0.0%) 非农户籍 88(54.3%) 62(38.3%) 12(7.4%) 0(0.0%) 0(0.0%) 农业户籍 58(58.6%) 30(30.3%) 11(11.1%) 0(0.0%) 0(0.0%) 未住校 51(56.7%) 28(31.1%) 11(12.2%) 0(0.0%) 0(0.0%) 住校1~3年 47(49.5%) 41(43.2%) 7(7.4%) 0(0.0%) 0(0.0%) 住校4年及以上 48(63.2%) 23(30.3%) 5(6.6%) 0(0.0%) 0(0.0%) 注：括号内数值为频数相应的比重. 
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表 2 地方感尺度偏好的人口统计学特征卡方检验参数
Table 2 The parameters of the chi-square test of demographic characteristics
特征 地方感 x2 Sig 性别 地方依附 3.667 0.454 地方认同 1.612 0.447 年龄 地方依附 17.160 0.002** 地方认同 4.057 0.401 独生/非独 地方依附 15.910 0.003** 地方认同 15.243 0.000** 民族 地方依附 21.101 0.000** 地方认同 7.012 0.023* 户籍 地方依附 33.273 0.000** 地方认同 2.263 0.323 专业 地方依附 14.290 0.027* 地方认同 2.621 0.863 年级 地方依附 3.988 0.136 地方认同 0.447 0.800 上大学前的住校时间 地方依附 12.737 0.013* 地方认同 5.928 0.205 注：*5%显著水平；**1%极显著水平.
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表 3 地方依附编码一致性检验结果
Table 3 The results of inter-rater reliability for coding of place attachment
代码 Kappa指数 主体 0.87 个人 0.89 群体 0.41 过程 0.87 情感 - 认知 0.83 行为 0.90 客体 0.90 地方的自然属性 - 地方的社会属性 0.81 地方的人文经济属性 0.89 地方的空间属性 0.75 其他 0.71 特定语境 - 未分类 -
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表 4 地方认同编码一致性检验结果
Table 4 The results of inter-rater reliability for coding of place identity
代码 Kappa指数 主体 0.89 个人 0.89 群体 0.48 过程 0.87 情感 - 认知 0.55 行为 0.61 客体 0.96 地方的自然属性 - 地方的社会属性 0.43 地方的人文经济属性 0.97 地方的空间属性 0.96 其他 0.56 特定语境 - 未分类 -
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表 5 被试地方感编码结果的频数分布
Table 5 The frequency distribution of coding results of subjects' sense of place
一级代码 二级代码 三级代码 地方依附 地方认同 频数/次 比重/% 频数/次 比重/% 主体 个人 经历 104 39.8 15 5.7 重大事件 25 9.6 3 1.1 自我实现 30 11.5 3 1.1 群体 地缘认同 28 10.7 19 7.3 民族认同 3 1.1 1 0.4 过程 情感 积极的情感 99 37.9 17 6.5 认知 熟悉 37 14.2 2 0.8 回忆与牵挂 16 6.1 1 0.4 观点 9 3.4 7 2.7 行为 习惯 17 6.5 58 22.2 承担义务 2 0.8 2 0.8 靠近家乡 4 1.5 1 0.4 客体 社会属性 社会符号 34 13.0 13 5.0 社交场合 19 7.3 1 0.4 人文属性 地方优势 39 14.9 121 46.4 地方的独特性 29 11.1 97 37.2 自然环境 自然环境 9 3.4 0 0.0 空间属性 尺度 45 17.2 42 16.1 距离 21 8.0 51 19.5 其他 其他 特定语境 5 1.9 25 9.6 未分类 6 2.3 12 4.6 总计 582 229.99 491 188.12
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