温莎日记 20

Milton Friedman - Capitalism and Freedom：

Main Contents:

1. 永久收入假說-弗里德曼1957年闡述: 為了未來勞動收入下降的儲蓄; 未雨綢繆式的儲蓄; 缺乏耐心而停止儲蓄.

2. 跨期最優化框架-假設是平方的效用; 假設公平受主管折扣費率和利率影響.

3. 消費經濟學文獻-預防性措施下的儲蓄; 流動性約束.

4. Bewley Models - 同質化的代理人廣義均衡不完全市場; 沒耐心的不儲蓄.

5. 未雨綢繆的函數- Caballero 1990-1991.

6. 這裡主要展示三個知識點: PIH是一種在模型4中化的規則, 從而保證每一個代理人都能解決模型5的問題; 持續性的未雨綢繆儲蓄需求函數和持續性的無耐心的不儲蓄、並存; 兩股力量, 彼此消耗在對方那裡的均衡勢力.

Model Setup:

Fix a probability space $(\Omega , F, P)$ and an information filtration $\left\{ F_t \right\} _{t=0}^∞$ .

(1) A stochastic uninsurable autoregressive income process: $y_t=\phi _0+\phi _1 y_{t-1} +\sigma w_t$ .

(2) Budget constant: $A_{t+1} =(1+r)A_t + y_t-c_t$ , $A_0$ given.

(3) Time-additive state-separable constant-absolute-risk-averse exponential utility:

$u(c)=-e^\left\{ - \theta c\right\} / \theta$ , $\theta >0$ ; $U(c)=E(\sum_{t=0}^∞ \left\{ \frac{1}{1+\delta } \right\} ^t u(c_t) )$ , $\delta >0$ .

(4) Optimal Consumption:

Bellman equation, $V(A_t,y_t)=sup_{c_t} \left\{ u(c_t)+(1/(1+\delta ))E_tV(A_{t+1},y_{t+1}) \right\}$ .

Conjecture of value function, $V(A,y)=-1/(\theta r)exp\left\{ -\theta r(A+ay+\tilde{b} ) \right\}$ .

Candidate optimal consumption, $c_t^*=r(A_t+ay_t+a_0)$ , $a_0=\tilde{b}+\frac{1}{\theta r} log(1+r)$ .

The Bellman model implies,

$V(A_t,y_t)=-\frac{r}{1+r} V(A_t,y_t)-(\frac{1}{1+\delta } \frac{1}{\theta r}E_texp[-\theta r(A_{t+1}+ay_{t+1}+\tilde{b} )] )$ .

Theorem. Suppose that the Laplace transformation $\zeta (\cdot )$ of the income innovation $w_t$ is finite over the range from 0 through $- \theta \sigma ra$ . The agent's optimal consumption rule for (4) is then

(5) $c_t^*=r(A_t +ay_t+a_0)$ .

(6) $a_0=\frac{a\phi _0}{r} -\Gamma (r)$ .

(7) $\Gamma (r)=\frac{1}{\theta r^2} [\Pi (r)-\Psi (r)]$ .

(8) $\Psi (r)=log(\frac{1+\delta }{1+r} )$ .

(9) $\Pi (r)=m(-\theta \sigma ra)$ .

(10) $a=\frac{1}{1+r-\phi _1}$ and $m(k)=log \zeta (k)$ .

(11) Friedman 1957 calculus, $h_t\equiv (\frac{1}{1+r} )E_t[\sum_{j=0}^∞\left\{ (\frac{1}{1+r} )^jy_{t+j} \right\} ]=a(y_t+\frac{\phi _0}{r} )$ .

(12) The optimal consumption gives the equality, $c_t^*=r(A_t+h_t-\Gamma (r))$ .

(13) The saving rate, $s_t^*=A_{t+1}^*-A_{t}^*=rA_{t}^*+y_t-c_t^*=f_t+\frac{1}{\theta r} \Pi (r)-\frac{1}{\theta r} \Psi (r)$ .

(14) The main component of (13) is, $f_t=a(1-\phi _1)(y_t-\tilde{y} )$ , and $\tilde{y}=\frac{\phi _0}{1-\phi _1}$ .

(15) Equilibrium of agent's consumption at rate $r^*\in (0,\delta )$ , in that $c_t^*=r^*(A_t+h_t)$ .

Inference of the CRRA-utility-based Bewley models:

1. Smaller $\sigma _y$ gives a lower precautionary saving $\rightarrow$ the equilibrium interest rate is closer to the subjective discount rate $\delta$ ;

2. It shows that the permanent-income hypothesis gives the optimal consumption rule.

3. Equilibrium makes the agent effectively impatient: impatience offsets precaution.

温莎日记 20

温莎日记 20

Milton Friedman - Capitalism and Freedom：

Main Contents:

Model Setup:

Inference of the CRRA-utility-based Bewley models:

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