货币理论与政策解决方案手册,第 4 版,麻省理工学院出版社 2017 年*
2017 年 3 月
内容
1 引言 ...1 2 第 2 章:Money-in-the-Utility 函数......2 3 第 3 章:货币和交易......15 4 第 4 章:货币与公共财政......35 5 第 5 章:信息和投资组合的刚性......46 6 第 6 章:自由裁量权政策和时间不一致......52 7 第七章:名义价格和工资刚性 ...82 8 第 8 章:新凯恩斯主义货币经济学 ...98 9 第 9 章:开放经济中的货币政策 ...127 10 第 10 章:金融市场和货币政策 ...146 11 第 11 章:有效的下限和资产负债表政策......163 12 第 12 章:货币政策操作程序 ...175
1 引言
2 第 2 章:Money-in-the-Utility 函数
第 2.2 节的 MIU 模型意味着货币和消费之间的边际替代率被设定为等于
i
t
/
(
1
+
i
t
)
i
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/
1
+
i
t
i_(t)//(1+i_(t)) i_{t} /\left(1+i_{t}\right) (见 (2.12))。该模型假设代理人带着资源进入时期
t
t
t t ,并使用这些资源
ω
t
ω
t
omega_(t) \omega_{t} 来购买资本、消费、名义债券和货币。这些货币持有的实际价值在期间
t
t
t t 产生了效用。相反,假设在 period
t
t
t t 中选择的货币持有量在 period
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+
1
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t+1 t+1 之前不会产生效用。效用和
∑
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sumbeta^(i)U(c_(t+i),M_(t+i)//P_(t+i)) \sum \beta^{i} U\left(c_{t+i}, M_{t+i} / P_{t+i}\right) 以前一样,但预算限制采用
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=
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+
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omega_(t)=c_(t)+(M_(t+1))/(P_(t))+b_(t)+k_(t) \omega_{t}=c_{t}+\frac{M_{t+1}}{P_{t}}+b_{t}+k_{t}
和家庭选择
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,
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c
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,
b
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c_(t),k_(t),b_(t) c_{t}, k_{t}, b_{t} ,并在
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+
1
M
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+
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M_(t+1) M_{t+1} 时期
t
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t t 。家庭的真正财富
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ω
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omega_(t) \omega_{t} 是由
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+
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omega_(t)=f(k_(t-1))+(1-delta)k_(t-1)+(1+r_(t-1))b_(t-1)+m_(t) \omega_{t}=f\left(k_{t-1}\right)+(1-\delta) k_{t-1}+\left(1+r_{t-1}\right) b_{t-1}+m_{t}
推导住户选择的一阶条件,
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M
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+
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M_(t+1) M_{t+1} 并证明
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=
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=
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(U_(m)(c_(t+1),m_(t+1)))/(U_(c)(c_(t+1),m_(t+1)))=i_(t) \frac{U_{m}\left(c_{t+1}, m_{t+1}\right)}{U_{c}\left(c_{t+1}, m_{t+1}\right)}=i_{t}
(由 Kevin Salyer 建议。设 value 函数为
V
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=
max
{
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+
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}
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+
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V(omega_(t),m_(t))=max{U(c_(t),m_(t))+beta V(omega_(t+1),m_(t+1))} V\left(\omega_{t}, m_{t}\right)=\max \left\{U\left(c_{t}, m_{t}\right)+\beta V\left(\omega_{t+1}, m_{t+1}\right)\right\}
受制于
ω
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+
1
=
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)
+
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)
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+
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+
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=
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)
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+
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omega_(t+1)=f(k_(t))+(1-delta)k_(t)+(1+r_(t))b_(t)+m_(t+1) \omega_{t+1}=f\left(k_{t}\right)+(1-\delta) k_{t}+\left(1+r_{t}\right) b_{t}+m_{t+1}
和
ω
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−
c
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−
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+
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+
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+
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−
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omega_(t)-c_(t)-m_(t+1)(1+pi_(t+1))-b_(t)-k_(t)=0 \omega_{t}-c_{t}-m_{t+1}\left(1+\pi_{t+1}\right)-b_{t}-k_{t}=0
在这种设置中,
r
t
r
t
r_(t) r_{t} 是债券的实际回报。让我们
λ
t
λ
t
lambda_(t) \lambda_{t} 表示最后一个约束的拉格朗日。
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+
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c_(t),m_(t+1),b_(t) c_{t}, m_{t+1}, b_{t} 的一阶条件 和
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k
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k_(t) k_{t} 加上包络定理得到
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=
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β
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+
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=
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=
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=
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=
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)
.
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=
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+
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m
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+
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+
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ω
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+
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=
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+
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+
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V
ω
ω
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+
1
,
m
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+
1
=
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V
ω
ω
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V
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ω
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,
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=
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c
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.
{:[U_(c)(c_(t),m_(t))=lambda_(t)],[betaV_(m)(omega_(t+1),m_(t+1))+betaV_(omega)(omega_(t+1),m_(t+1))=lambda_(t)(1+pi_(t+1))],[beta(1+r_(t))V_(omega)(omega_(t+1),m_(t+1))=lambda_(t)],[V_(omega)(omega_(t),m_(t))=lambda_(t)],[V_(m)(omega_(t),m_(t))=U_(m)(c_(t),m_(t)).]:} \begin{gathered}
U_{c}\left(c_{t}, m_{t}\right)=\lambda_{t} \\
\beta V_{m}\left(\omega_{t+1}, m_{t+1}\right)+\beta V_{\omega}\left(\omega_{t+1}, m_{t+1}\right)=\lambda_{t}\left(1+\pi_{t+1}\right) \\
\beta\left(1+r_{t}\right) V_{\omega}\left(\omega_{t+1}, m_{t+1}\right)=\lambda_{t} \\
V_{\omega}\left(\omega_{t}, m_{t}\right)=\lambda_{t} \\
V_{m}\left(\omega_{t}, m_{t}\right)=U_{m}\left(c_{t}, m_{t}\right) .
\end{gathered}
现在梳理其中的第二个和第三个以获得
β
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+
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=
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V
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β
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+
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+
β
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ω
ω
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+
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=
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+
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V
ω
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+
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betaV_(m)(omega_(t+1),m_(t+1))+betaV_(omega)(omega_(t+1),m_(t+1))=beta(1+pi_(t+1))(1+r_(t))V_(omega)(omega_(t+1),m_(t+1)) \beta V_{m}\left(\omega_{t+1}, m_{t+1}\right)+\beta V_{\omega}\left(\omega_{t+1}, m_{t+1}\right)=\beta\left(1+\pi_{t+1}\right)\left(1+r_{t}\right) V_{\omega}\left(\omega_{t+1}, m_{t+1}\right)
或
β
V
m
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+
1
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+
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)
=
[
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+
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−
1
]
β
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ω
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=
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)
.
β
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ω
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=
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1
+
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β
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ω
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=
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β
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ω
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,
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+
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.
{:[betaV_(m)(omega_(t+1),m_(t+1))=[(1+pi_(t+1))(1+r_(t))-1]betaV_(omega)(omega_(t+1),m_(t+1))],[=i_(t)betaV_(omega)(omega_(t+1),m_(t+1)).]:} \begin{aligned}
\beta V_{m}\left(\omega_{t+1}, m_{t+1}\right) & =\left[\left(1+\pi_{t+1}\right)\left(1+r_{t}\right)-1\right] \beta V_{\omega}\left(\omega_{t+1}, m_{t+1}\right) \\
& =i_{t} \beta V_{\omega}\left(\omega_{t+1}, m_{t+1}\right) .
\end{aligned}
因此
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V
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=
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(V_(m)(omega_(t+1),m_(t+1)))/(V_(omega)(omega_(t+1),m_(t+1)))=i_(t) \frac{V_{m}\left(\omega_{t+1}, m_{t+1}\right)}{V_{\omega}\left(\omega_{t+1}, m_{t+1}\right)}=i_{t}
但是最后一个一阶条件暗示
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=
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V_(m)(omega_(t+1),m_(t+1))=U_(m)(c_(t+1),m_(t+1)) V_{m}\left(\omega_{t+1}, m_{t+1}\right)=U_{m}\left(c_{t+1}, m_{t+1}\right) ,而第一个和第四个产生
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=
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=
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V
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=
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=
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V_(omega)(omega_(t+1),m_(t+1))=lambda_(t+1)=U_(c)(c_(t+1),m_(t+1)) V_{\omega}\left(\omega_{t+1}, m_{t+1}\right)=\lambda_{t+1}=U_{c}\left(c_{t+1}, m_{t+1}\right) 。因此
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+
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,
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)
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=
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=
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(U_(m)(c_(t+1),m_(t+1)))/(U_(c)(c_(t+1),m_(t+1)))=i_(t) \frac{U_{m}\left(c_{t+1}, m_{t+1}\right)}{U_{c}\left(c_{t+1}, m_{t+1}\right)}=i_{t}
在第 2.2 节考虑的案例中,当时
t
t
t t 持有的现金在当时
t
t
t t 产生了效用,但机会成本在于本可以持有的债券的利息收入损失。由于这笔利息支付发生在
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+
1
t
+
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t+1 t+1 ,因此必须将其贴现回去,以与当时持有货币的边际价值进行比较
t
t
t t 。 2. (Carlstrom 和 Fuerst (2001)。假设代表住户的效用取决于消费和可用于消费的实际货币余额水平。设
A
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/
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A
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A_(t)//P_(t) A_{t} / P_{t} 为进入效用函数的真实货币存量。如果忽略资本,则家庭的目标是在预算约束下实现最大化
∑
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sumbeta^(i)U(c_(t+i),A_(t+i)//P_(t+i)) \sum \beta^{i} U\left(c_{t+i}, A_{t+i} / P_{t+i}\right)
Y
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+
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,
Y_(t)+(M_(t-5))/(P_(t))+tau_(t)+((1+i_(t-1))B_(t尺1))/(P_(t))=C_(t)+(M_(t))/(P_(t))+(B_(t))/(P_(t)), Y_{t}+\frac{M_{t-5}}{P_{t}}+\tau_{t}+\frac{\left(1+i_{t-1}\right) B_{t 尺 1}}{P_{t}}=C_{t}+\frac{M_{t}}{P_{t}}+\frac{B_{t}}{P_{t}}, 尺
其中收入
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t
Y
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Y_(t) Y_{t} 被视为一个外生过程。假设产生效用的货币存量是购买债券之后但在收到收入或购买消费品之前持有的货币的实际价值:
A
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A
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−
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(A_(t))/(P_(t))=(M_(t-1))/(P_(t))+tau_(t)+((1+i_(t-1))B_(t-1))/(P_(t))-(B_(t))/(P_(t)) \frac{A_{t}}{P_{t}}=\frac{M_{t-1}}{P_{t}}+\tau_{t}+\frac{\left(1+i_{t-1}\right) B_{t-1}}{P_{t}}-\frac{B_{t}}{P_{t}}
(a) 推导
B
t
B
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B_(t) B_{t} 和 的
A
t
A
t
A_(t) A_{t} 一阶条件。设 value 函数为
V
(
z
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)
=
max
{
U
(
c
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,
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−
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)
+
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V
(
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+
1
}
,
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t
=
max
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c
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,
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−
b
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+
β
V
z
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+
1
,
V(z_(t))=max{U(c_(t),z_(t)-b_(t))+beta V(z_(t+1)},:} V\left(z_{t}\right)=\max \left\{U\left(c_{t}, z_{t}-b_{t}\right)+\beta V\left(z_{t+1}\right\},\right.
哪里
z
t
+
1
=
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P
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+
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+
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+
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+
(
1
+
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)
B
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P
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+
1
=
(
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+
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)
m
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+
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+
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+
(
1
+
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)
(
P
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P
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+
1
)
b
t
z
t
+
1
=
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t
P
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+
1
+
τ
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+
1
+
1
+
i
t
B
t
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+
1
=
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+
1
m
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+
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+
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+
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P
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+
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b
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{:[z_(t+1)=(M_(t))/(P_(t+1))+tau_(t+1)+((1+i_(t))B_(t))/(P_(t+1))],[=((P_(t))/(P_(t+1)))m_(t)+tau_(t+1)+(1+i_(t))((P_(t))/(P_(t+1)))b_(t)]:} \begin{aligned}
z_{t+1} & =\frac{M_{t}}{P_{t+1}}+\tau_{t+1}+\frac{\left(1+i_{t}\right) B_{t}}{P_{t+1}} \\
& =\left(\frac{P_{t}}{P_{t+1}}\right) m_{t}+\tau_{t+1}+\left(1+i_{t}\right)\left(\frac{P_{t}}{P_{t+1}}\right) b_{t}
\end{aligned}
和
Y
t
+
z
t
−
c
t
−
m
t
−
b
t
=
0
Y
t
+
z
t
−
c
t
−
m
t
−
b
t
=
0
Y_(t)+z_(t)-c_(t)-m_(t)-b_(t)=0 Y_{t}+z_{t}-c_{t}-m_{t}-b_{t}=0
在时间
t
t
t t ,家庭选择
c
t
,
m
t
,
b
t
c
t
,
m
t
,
b
t
c_(t),m_(t),b_(t) c_{t}, m_{t}, b_{t} 。让我们
λ
t
λ
t
lambda_(t) \lambda_{t} 表示最后一个约束的拉格朗日。一阶条件加上包络定理得到
U
c
(
c
t
,
z
t
−
b
t
)
=
λ
t
β
(
P
t
P
t
+
1
)
V
z
(
z
t
+
1
)
=
λ
t
−
U
a
(
c
t
,
a
t
)
+
β
(
1
+
i
t
)
(
P
t
P
t
+
1
)
V
z
(
z
t
+
1
)
=
λ
t
V
z
(
z
t
)
=
U
a
(
c
t
,
z
t
−
b
t
)
+
λ
t
U
c
c
t
,
z
t
−
b
t
=
λ
t
β
P
t
P
t
+
1
V
z
z
t
+
1
=
λ
t
−
U
a
c
t
,
a
t
+
β
1
+
i
t
P
t
P
t
+
1
V
z
z
t
+
1
=
λ
t
V
z
z
t
=
U
a
c
t
,
z
t
−
b
t
+
λ
t
{:[U_(c)(c_(t),z_(t)-b_(t))=lambda_(t)],[beta((P_(t))/(P_(t+1)))V_(z)(z_(t+1))=lambda_(t)],[-U_(a)(c_(t),a_(t))+beta(1+i_(t))((P_(t))/(P_(t+1)))V_(z)(z_(t+1))=lambda_(t)],[V_(z)(z_(t))=U_(a)(c_(t),z_(t)-b_(t))+lambda_(t)]:} \begin{gathered}
U_{c}\left(c_{t}, z_{t}-b_{t}\right)=\lambda_{t} \\
\beta\left(\frac{P_{t}}{P_{t+1}}\right) V_{z}\left(z_{t+1}\right)=\lambda_{t} \\
-U_{a}\left(c_{t}, a_{t}\right)+\beta\left(1+i_{t}\right)\left(\frac{P_{t}}{P_{t+1}}\right) V_{z}\left(z_{t+1}\right)=\lambda_{t} \\
V_{z}\left(z_{t}\right)=U_{a}\left(c_{t}, z_{t}-b_{t}\right)+\lambda_{t}
\end{gathered}
现在将其中的第二个和第三个组合在一起以消除 value 函数,可以得到
−
U
a
(
c
t
,
z
t
−
b
t
)
+
(
1
+
i
t
)
λ
t
=
λ
t
−
U
a
c
t
,
z
t
−
b
t
+
1
+
i
t
λ
t
=
λ
t
-U_(a)(c_(t),z_(t)-b_(t))+(1+i_(t))lambda_(t)=lambda_(t) -U_{a}\left(c_{t}, z_{t}-b_{t}\right)+\left(1+i_{t}\right) \lambda_{t}=\lambda_{t} ,或
U
a
(
c
t
,
a
t
)
=
i
t
λ
t
.
U
a
c
t
,
a
t
=
i
t
λ
t
.
U_(a)(c_(t),a_(t))=i_(t)lambda_(t). U_{a}\left(c_{t}, a_{t}\right)=i_{t} \lambda_{t} .
在第四个方程 中使用它,
V
z
(
z
t
)
=
U
a
(
c
t
,
a
t
)
+
λ
t
=
λ
t
(
1
+
i
t
)
V
z
z
t
=
U
a
c
t
,
a
t
+
λ
t
=
λ
t
1
+
i
t
V_(z)(z_(t))=U_(a)(c_(t),a_(t))+lambda_(t)=lambda_(t)(1+i_(t)) V_{z}\left(z_{t}\right)=U_{a}\left(c_{t}, a_{t}\right)+\lambda_{t}=\lambda_{t}\left(1+i_{t}\right) .现在更新这一时期并使用它来消除
V
z
(
z
t
+
1
)
V
z
z
t
+
1
V_(z)(z_(t+1)) V_{z}\left(z_{t+1}\right) 第二个
F
O
C
F
O
C
FOC F O C 时期,我们只剩下
U
c
(
c
t
,
a
t
)
=
λ
t
λ
t
=
β
(
1
+
i
t
+
1
)
(
P
t
P
t
+
1
)
λ
t
+
1
=
β
(
1
+
i
t
+
1
1
+
π
t
+
1
)
λ
t
+
1
U
c
c
t
,
a
t
=
λ
t
λ
t
=
β
1
+
i
t
+
1
P
t
P
t
+
1
λ
t
+
1
=
β
1
+
i
t
+
1
1
+
π
t
+
1
λ
t
+
1
{:[U_(c)(c_(t),a_(t))=lambda_(t)],[lambda_(t)=beta(1+i_(t+1))((P_(t))/(P_(t+1)))lambda_(t+1)=beta((1+i_(t+1))/(1+pi_(t+1)))lambda_(t+1)]:} \begin{gathered}
U_{c}\left(c_{t}, a_{t}\right)=\lambda_{t} \\
\lambda_{t}=\beta\left(1+i_{t+1}\right)\left(\frac{P_{t}}{P_{t+1}}\right) \lambda_{t+1}=\beta\left(\frac{1+i_{t+1}}{1+\pi_{t+1}}\right) \lambda_{t+1}
\end{gathered}
(乙)这些条件跟经文里的条件有什么不同?在第 2.2 节的基本 MIU 模型中,方程 (2.10) 和 (1) 相同。使用 (2.7) 和 (2.10),方程 (2.6) 可以写成(设置
n
=
0
n
=
0
n=0 n=0 后)
TESTB
λ
t
I
∄
β
(
1
+
i
t
1
+
π
t
+
1
)
λ
t
+
1
TESTB
λ
t
I
∄
β
1
+
i
t
1
+
π
t
+
1
λ
t
+
1
TESTBlambda_(t)I∄beta((1+i_(t))/(1+pi_(t+1)))lambda_(t+1) \mathrm{TESTB} \lambda_{t} I \nexists \beta\left(\frac{1+i_{t}}{1+\pi_{t+1}}\right) \lambda_{t+1}
which differs from (2) in that
i
t
i
t
i_(t) i_{t} rather than
i
t
+
1
i
t
+
1
i_(t+1) i_{t+1} appears. This difference in timing is related to the dependence of utility on
M
t
/
P
t
M
t
/
P
t
M_(t)//P_(t) M_{t} / P_{t} in the model of the text and on
A
t
/
P
t
=
A
t
/
P
t
=
A_(t)//P_(t)= A_{t} / P_{t}=
M
t
−
1
/
P
t
+
τ
t
+
[
(
1
+
i
t
−
1
)
B
t
−
1
−
B
t
]
/
P
t
M
t
−
1
/
P
t
+
τ
t
+
1
+
i
t
−
1
B
t
−
1
−
B
t
/
P
t
M_(t-1)//P_(t)+tau_(t)+[(1+i_(t-1))B_(t-1)-B_(t)]//P_(t) M_{t-1} / P_{t}+\tau_{t}+\left[\left(1+i_{t-1}\right) B_{t-1}-B_{t}\right] / P_{t} in the model of this problem. In the basic MIU model of section 2.2, the household can reduce current consumption at a cost of
U
c
(
c
t
,
m
t
)
=
λ
t
U
c
c
t
,
m
t
=
λ
t
U_(c)(c_(t),m_(t))=lambda_(t) U_{c}\left(c_{t}, m_{t}\right)=\lambda_{t} and purchase bonds instead, yielding a real payoff of
(
1
+
i
t
)
/
(
1
+
π
t
+
1
)
1
+
i
t
/
1
+
π
t
+
1
(1+i_(t))//(1+pi_(t+1)) \left(1+i_{t}\right) /\left(1+\pi_{t+1}\right) in period
t
+
1
t
+
1
t+1 t+1 . This gross interest income could be spend on consumption, but it did not serve to augment the period
t
+
1
t
+
1
t+1 t+1 holdings of money that enter the utility function. In the model of this problem, the same option to lower current consumption to purchase bonds is open, but purchasing bonds has an addition cost since purchasing additional bonds reduces
A
t
/
P
t
A
t
/
P
t
A_(t)//P_(t) A_{t} / P_{t} which reduces current utility by
U
a
(
c
t
,
z
t
−
b
t
)
=
i
t
λ
t
U
a
c
t
,
z
t
−
b
t
=
i
t
λ
t
U_(a)(c_(t),z_(t)-b_(t))=i_(t)lambda_(t) U_{a}\left(c_{t}, z_{t}-b_{t}\right)=i_{t} \lambda_{t} in period
t
t
t t . Thus, the utility cost of purchasing a bond is
U
c
(
c
t
,
z
t
−
b
t
)
+
U
a
(
c
t
,
z
t
−
b
t
)
=
(
1
+
i
t
)
λ
t
U
c
c
t
,
z
t
−
b
t
+
U
a
c
t
,
z
t
−
b
t
=
1
+
i
t
λ
t
U_(c)(c_(t),z_(t)-b_(t))+U_(a)(c_(t),z_(t)-b_(t))=(1+i_(t))lambda_(t) U_{c}\left(c_{t}, z_{t}-b_{t}\right)+U_{a}\left(c_{t}, z_{t}-b_{t}\right)=\left(1+i_{t}\right) \lambda_{t} . The bond pays off
1
+
i
t
)
/
(
1
+
π
t
+
1
)
1
+
i
t
/
1
+
π
t
+
1
{:1+i_(t))//(1+pi_(t+1)) \left.1+i_{t}\right) /\left(1+\pi_{t+1}\right) in period
t
+
1
t
+
1
t+1 t+1 and this return, since it is paid out before the goods market opens, yields the marginal utility of
A
t
+
1
/
P
t
+
1
A
t
+
1
/
P
t
+
1
A_(t+1)//P_(t+1) A_{t+1} / P_{t+1} (equal to
i
t
+
1
λ
t
+
1
i
t
+
1
λ
t
+
1
i_(t+1)lambda_(t+1) i_{t+1} \lambda_{t+1} ) plus the marginal utility of consumption
(
λ
t
+
1
)
λ
t
+
1
(lambda_(t+1)) \left(\lambda_{t+1}\right) . That is, the gross interest income from the bond augments the real stock of money that enters time
t
t
t t utility and can be spend on period
t
t
t t consumption. Hence,
(
1
+
i
t
)
λ
t
=
β
(
1
+
i
t
1
+
π
t
+
1
)
(
1
+
i
t
+
1
)
λ
t
+
1
⇒
λ
t
=
β
(
1
+
i
t
+
1
1
+
π
t
+
1
)
λ
t
+
1
1
+
i
t
λ
t
=
β
1
+
i
t
1
+
π
t
+
1
1
+
i
t
+
1
λ
t
+
1
⇒
λ
t
=
β
1
+
i
t
+
1
1
+
π
t
+
1
λ
t
+
1
(1+i_(t))lambda_(t)=beta((1+i_(t))/(1+pi_(t+1)))(1+i_(t+1))lambda_(t+1)=>lambda_(t)=beta((1+i_(t+1))/(1+pi_(t+1)))lambda_(t+1) \left(1+i_{t}\right) \lambda_{t}=\beta\left(\frac{1+i_{t}}{1+\pi_{t+1}}\right)\left(1+i_{t+1}\right) \lambda_{t+1} \Rightarrow \lambda_{t}=\beta\left(\frac{1+i_{t+1}}{1+\pi_{t+1}}\right) \lambda_{t+1}